1.13 quiz problem solving with scientific notation

Scientific Notation Quiz

This online quiz is intended to give you extra practice in converting between scientific notation and decimal notation.

Select your preferences below and click 'Start' to give it a try!

1.2 Exponents and Scientific Notation

Learning objectives.

In this section, you will:

• Use the product rule of exponents.
• Use the quotient rule of exponents.
• Use the power rule of exponents.
• Use the zero exponent rule of exponents.
• Use the negative rule of exponents.
• Find the power of a product and a quotient.
• Simplify exponential expressions.
• Use scientific notation.

Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.

Using a calculator, we enter 2,048 × 1,536 × 48 × 24 × 3,600 2,048 × 1,536 × 48 × 24 × 3,600 and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of approximately 1.3 × 10 13 1.3 × 10 13 bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers.

Using the Product Rule of Exponents

Consider the product x 3 ⋅ x 4 . x 3 ⋅ x 4 . Both terms have the same base, x , but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

The result is that x 3 ⋅ x 4 = x 3 + 4 = x 7 . x 3 ⋅ x 4 = x 3 + 4 = x 7 .

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

Now consider an example with real numbers.

We can always check that this is true by simplifying each exponential expression. We find that 2 3 2 3 is 8, 2 4 2 4 is 16, and 2 7 2 7 is 128. The product 8 ⋅ 16 8 ⋅ 16 equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

The Product Rule of Exponents

For any real number a a and natural numbers m m and n , n , the product rule of exponents states that

Using the Product Rule

Write each of the following products with a single base. Do not simplify further.

• ⓐ t 5 ⋅ t 3 t 5 ⋅ t 3
• ⓑ ( −3 ) 5 ⋅ ( −3 ) ( −3 ) 5 ⋅ ( −3 )
• ⓒ x 2 ⋅ x 5 ⋅ x 3 x 2 ⋅ x 5 ⋅ x 3

Use the product rule to simplify each expression.

• ⓐ t 5 ⋅ t 3 = t 5 + 3 = t 8 t 5 ⋅ t 3 = t 5 + 3 = t 8
• ⓑ ( −3 ) 5 ⋅ ( −3 ) = ( −3 ) 5 ⋅ ( −3 ) 1 = ( −3 ) 5 + 1 = ( −3 ) 6 ( −3 ) 5 ⋅ ( −3 ) = ( −3 ) 5 ⋅ ( −3 ) 1 = ( −3 ) 5 + 1 = ( −3 ) 6

At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.

Notice we get the same result by adding the three exponents in one step.

• ⓐ k 6 ⋅ k 9 k 6 ⋅ k 9
• ⓑ ( 2 y ) 4 ⋅ ( 2 y ) ( 2 y ) 4 ⋅ ( 2 y )
• ⓒ t 3 ⋅ t 6 ⋅ t 5 t 3 ⋅ t 6 ⋅ t 5

Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as y m y n , y m y n , where m > n . m > n . Consider the example y 9 y 5 . y 9 y 5 . Perform the division by canceling common factors.

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.

In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

For the time being, we must be aware of the condition m > n . m > n . Otherwise, the difference m − n m − n could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

The Quotient Rule of Exponents

For any real number a a and natural numbers m m and n , n , such that m > n , m > n , the quotient rule of exponents states that

Using the Quotient Rule

• ⓐ ( −2 ) 14 ( −2 ) 9 ( −2 ) 14 ( −2 ) 9
• ⓑ t 23 t 15 t 23 t 15
• ⓒ ( z 2 ) 5 z 2 ( z 2 ) 5 z 2

Use the quotient rule to simplify each expression.

• ⓐ ( −2 ) 14 ( −2 ) 9 = ( −2 ) 14 − 9 = ( −2 ) 5 ( −2 ) 14 ( −2 ) 9 = ( −2 ) 14 − 9 = ( −2 ) 5
• ⓑ t 23 t 15 = t 23 − 15 = t 8 t 23 t 15 = t 23 − 15 = t 8
• ⓒ ( z 2 ) 5 z 2 = ( z 2 ) 5 − 1 = ( z 2 ) 4 ( z 2 ) 5 z 2 = ( z 2 ) 5 − 1 = ( z 2 ) 4
• ⓐ s 75 s 68 s 75 s 68
• ⓑ ( −3 ) 6 −3 ( −3 ) 6 −3
• ⓒ ( e f 2 ) 5 ( e f 2 ) 3 ( e f 2 ) 5 ( e f 2 ) 3

Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents . Consider the expression ( x 2 ) 3 . ( x 2 ) 3 . The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.

The exponent of the answer is the product of the exponents: ( x 2 ) 3 = x 2 ⋅ 3 = x 6 . ( x 2 ) 3 = x 2 ⋅ 3 = x 6 . In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.

Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

The Power Rule of Exponents

For any real number a a and positive integers m m and n , n , the power rule of exponents states that

Using the Power Rule

• ⓐ ( x 2 ) 7 ( x 2 ) 7
• ⓑ ( ( 2 t ) 5 ) 3 ( ( 2 t ) 5 ) 3
• ⓒ ( ( −3 ) 5 ) 11 ( ( −3 ) 5 ) 11

Use the power rule to simplify each expression.

• ⓐ ( x 2 ) 7 = x 2 ⋅ 7 = x 14 ( x 2 ) 7 = x 2 ⋅ 7 = x 14
• ⓑ ( ( 2 t ) 5 ) 3 = ( 2 t ) 5 ⋅ 3 = ( 2 t ) 15 ( ( 2 t ) 5 ) 3 = ( 2 t ) 5 ⋅ 3 = ( 2 t ) 15
• ⓒ ( ( −3 ) 5 ) 11 = ( −3 ) 5 ⋅ 11 = ( −3 ) 55 ( ( −3 ) 5 ) 11 = ( −3 ) 5 ⋅ 11 = ( −3 ) 55
• ⓐ ( ( 3 y ) 8 ) 3 ( ( 3 y ) 8 ) 3
• ⓑ ( t 5 ) 7 ( t 5 ) 7
• ⓒ ( ( − g ) 4 ) 4 ( ( − g ) 4 ) 4

Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that m > n m > n so that the difference m − n m − n would never be zero or negative. What would happen if m = n ? m = n ? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example.

If we were to simplify the original expression using the quotient rule, we would have

If we equate the two answers, the result is t 0 = 1. t 0 = 1. This is true for any nonzero real number, or any variable representing a real number.

The sole exception is the expression 0 0 . 0 0 . This appears later in more advanced courses, but for now, we will consider the value to be undefined.

The Zero Exponent Rule of Exponents

For any nonzero real number a , a , the zero exponent rule of exponents states that

Using the Zero Exponent Rule

Simplify each expression using the zero exponent rule of exponents.

• ⓐ c 3 c 3 c 3 c 3
• ⓑ −3 x 5 x 5 −3 x 5 x 5
• ⓒ ( j 2 k ) 4 ( j 2 k ) ⋅ ( j 2 k ) 3 ( j 2 k ) 4 ( j 2 k ) ⋅ ( j 2 k ) 3
• ⓓ 5 ( r s 2 ) 2 ( r s 2 ) 2 5 ( r s 2 ) 2 ( r s 2 ) 2

Use the zero exponent and other rules to simplify each expression.

ⓐ c 3 c 3 = c 3 − 3 = c 0 = 1 c 3 c 3 = c 3 − 3 = c 0 = 1

ⓑ −3 x 5 x 5 = −3 ⋅ x 5 x 5 = −3 ⋅ x 5 − 5 = −3 ⋅ x 0 = −3 ⋅ 1 = −3 −3 x 5 x 5 = −3 ⋅ x 5 x 5 = −3 ⋅ x 5 − 5 = −3 ⋅ x 0 = −3 ⋅ 1 = −3

ⓒ ( j 2 k ) 4 ( j 2 k ) ⋅ ( j 2 k ) 3 = ( j 2 k ) 4 ( j 2 k ) 1 + 3 Use the product rule in the denominator . = ( j 2 k ) 4 ( j 2 k ) 4 Simplify . = ( j 2 k ) 4 − 4 Use the quotient rule . = ( j 2 k ) 0 Simplify . = 1 ( j 2 k ) 4 ( j 2 k ) ⋅ ( j 2 k ) 3 = ( j 2 k ) 4 ( j 2 k ) 1 + 3 Use the product rule in the denominator . = ( j 2 k ) 4 ( j 2 k ) 4 Simplify . = ( j 2 k ) 4 − 4 Use the quotient rule . = ( j 2 k ) 0 Simplify . = 1

ⓓ 5 ( r s 2 ) 2 ( r s 2 ) 2 = 5 ( r s 2 ) 2 − 2 Use the quotient rule . = 5 ( r s 2 ) 0 Simplify . = 5 ⋅ 1 Use the zero exponent rule . = 5 Simplify . 5 ( r s 2 ) 2 ( r s 2 ) 2 = 5 ( r s 2 ) 2 − 2 Use the quotient rule . = 5 ( r s 2 ) 0 Simplify . = 5 ⋅ 1 Use the zero exponent rule . = 5 Simplify .

• ⓐ t 7 t 7 t 7 t 7
• ⓑ ( d e 2 ) 11 2 ( d e 2 ) 11 ( d e 2 ) 11 2 ( d e 2 ) 11
• ⓒ w 4 ⋅ w 2 w 6 w 4 ⋅ w 2 w 6
• ⓓ t 3 ⋅ t 4 t 2 ⋅ t 5 t 3 ⋅ t 4 t 2 ⋅ t 5

Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that m > n m > n in the quotient rule even further. For example, can we simplify h 3 h 5 ? h 3 h 5 ? When m < n m < n —that is, where the difference m − n m − n is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal.

Divide one exponential expression by another with a larger exponent. Use our example, h 3 h 5 . h 3 h 5 .

Putting the answers together, we have h −2 = 1 h 2 . h −2 = 1 h 2 . This is true for any nonzero real number, or any variable representing a nonzero real number.

A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.

We have shown that the exponential expression a n a n is defined when n n is a natural number, 0, or the negative of a natural number. That means that a n a n is defined for any integer n . n . Also, the product and quotient rules and all of the rules we will look at soon hold for any integer n . n .

The Negative Rule of Exponents

For any nonzero real number a a and natural number n , n , the negative rule of exponents states that

Using the Negative Exponent Rule

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

• ⓐ θ 3 θ 10 θ 3 θ 10
• ⓑ z 2 ⋅ z z 4 z 2 ⋅ z z 4
• ⓒ ( −5 t 3 ) 4 ( −5 t 3 ) 8 ( −5 t 3 ) 4 ( −5 t 3 ) 8
• ⓐ θ 3 θ 10 = θ 3 − 10 = θ −7 = 1 θ 7 θ 3 θ 10 = θ 3 − 10 = θ −7 = 1 θ 7
• ⓑ z 2 ⋅ z z 4 = z 2 + 1 z 4 = z 3 z 4 = z 3 − 4 = z −1 = 1 z z 2 ⋅ z z 4 = z 2 + 1 z 4 = z 3 z 4 = z 3 − 4 = z −1 = 1 z
• ⓒ ( −5 t 3 ) 4 ( −5 t 3 ) 8 = ( −5 t 3 ) 4 − 8 = ( −5 t 3 ) −4 = 1 ( −5 t 3 ) 4 ( −5 t 3 ) 4 ( −5 t 3 ) 8 = ( −5 t 3 ) 4 − 8 = ( −5 t 3 ) −4 = 1 ( −5 t 3 ) 4
• ⓐ ( −3 t ) 2 ( −3 t ) 8 ( −3 t ) 2 ( −3 t ) 8
• ⓑ f 47 f 49 ⋅ f f 47 f 49 ⋅ f
• ⓒ 2 k 4 5 k 7 2 k 4 5 k 7

Using the Product and Quotient Rules

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

• ⓐ b 2 ⋅ b −8 b 2 ⋅ b −8
• ⓑ ( − x ) 5 ⋅ ( − x ) −5 ( − x ) 5 ⋅ ( − x ) −5
• ⓒ −7 z ( −7 z ) 5 −7 z ( −7 z ) 5
• ⓐ b 2 ⋅ b −8 = b 2 − 8 = b −6 = 1 b 6 b 2 ⋅ b −8 = b 2 − 8 = b −6 = 1 b 6
• ⓑ ( − x ) 5 ⋅ ( − x ) −5 = ( − x ) 5 − 5 = ( − x ) 0 = 1 ( − x ) 5 ⋅ ( − x ) −5 = ( − x ) 5 − 5 = ( − x ) 0 = 1
• ⓒ −7 z ( −7 z ) 5 = ( −7 z ) 1 ( −7 z ) 5 = ( −7 z ) 1 − 5 = ( −7 z ) −4 = 1 ( −7 z ) 4 −7 z ( −7 z ) 5 = ( −7 z ) 1 ( −7 z ) 5 = ( −7 z ) 1 − 5 = ( −7 z ) −4 = 1 ( −7 z ) 4
• ⓐ t −11 ⋅ t 6 t −11 ⋅ t 6
• ⓑ 25 12 25 13 25 12 25 13

Finding the Power of a Product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider ( p q ) 3 . ( p q ) 3 . We begin by using the associative and commutative properties of multiplication to regroup the factors.

In other words, ( p q ) 3 = p 3 ⋅ q 3 . ( p q ) 3 = p 3 ⋅ q 3 .

The Power of a Product Rule of Exponents

For any real numbers a a and b b and any integer n , n , the power of a product rule of exponents states that

Using the Power of a Product Rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

• ⓐ ( a b 2 ) 3 ( a b 2 ) 3
• ⓑ ( 2 t ) 15 ( 2 t ) 15
• ⓒ ( −2 w 3 ) 3 ( −2 w 3 ) 3
• ⓓ 1 ( −7 z ) 4 1 ( −7 z ) 4
• ⓔ ( e −2 f 2 ) 7 ( e −2 f 2 ) 7

Use the product and quotient rules and the new definitions to simplify each expression.

• ⓐ ( a b 2 ) 3 = ( a ) 3 ⋅ ( b 2 ) 3 = a 1 ⋅ 3 ⋅ b 2 ⋅ 3 = a 3 b 6 ( a b 2 ) 3 = ( a ) 3 ⋅ ( b 2 ) 3 = a 1 ⋅ 3 ⋅ b 2 ⋅ 3 = a 3 b 6
• ⓑ ( 2 t ) 15 = ( 2 ) 15 ⋅ ( t ) 15 = 2 15 t 15 = 32 , 768 t 15 ( 2 t ) 15 = ( 2 ) 15 ⋅ ( t ) 15 = 2 15 t 15 = 32 , 768 t 15
• ⓒ ( −2 w 3 ) 3 = ( −2 ) 3 ⋅ ( w 3 ) 3 = −8 ⋅ w 3 ⋅ 3 = −8 w 9 ( −2 w 3 ) 3 = ( −2 ) 3 ⋅ ( w 3 ) 3 = −8 ⋅ w 3 ⋅ 3 = −8 w 9
• ⓓ 1 ( −7 z ) 4 = 1 ( −7 ) 4 ⋅ ( z ) 4 = 1 2 , 401 z 4 1 ( −7 z ) 4 = 1 ( −7 ) 4 ⋅ ( z ) 4 = 1 2 , 401 z 4
• ⓔ ( e −2 f 2 ) 7 = ( e −2 ) 7 ⋅ ( f 2 ) 7 = e −2 ⋅ 7 ⋅ f 2 ⋅ 7 = e −14 f 14 = f 14 e 14 ( e −2 f 2 ) 7 = ( e −2 ) 7 ⋅ ( f 2 ) 7 = e −2 ⋅ 7 ⋅ f 2 ⋅ 7 = e −14 f 14 = f 14 e 14
• ⓐ ( g 2 h 3 ) 5 ( g 2 h 3 ) 5
• ⓑ ( 5 t ) 3 ( 5 t ) 3
• ⓒ ( −3 y 5 ) 3 ( −3 y 5 ) 3
• ⓓ 1 ( a 6 b 7 ) 3 1 ( a 6 b 7 ) 3
• ⓔ ( r 3 s −2 ) 4 ( r 3 s −2 ) 4

Finding the Power of a Quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

Let’s rewrite the original problem differently and look at the result.

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

The Power of a Quotient Rule of Exponents

For any real numbers a a and b b and any integer n , n , the power of a quotient rule of exponents states that

Using the Power of a Quotient Rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

• ⓐ ( 4 z 11 ) 3 ( 4 z 11 ) 3
• ⓑ ( p q 3 ) 6 ( p q 3 ) 6
• ⓒ ( −1 t 2 ) 27 ( −1 t 2 ) 27
• ⓓ ( j 3 k −2 ) 4 ( j 3 k −2 ) 4
• ⓔ ( m −2 n −2 ) 3 ( m −2 n −2 ) 3
• ⓐ ( 4 z 11 ) 3 = ( 4 ) 3 ( z 11 ) 3 = 64 z 11 ⋅ 3 = 64 z 33 ( 4 z 11 ) 3 = ( 4 ) 3 ( z 11 ) 3 = 64 z 11 ⋅ 3 = 64 z 33
• ⓑ ( p q 3 ) 6 = ( p ) 6 ( q 3 ) 6 = p 1 ⋅ 6 q 3 ⋅ 6 = p 6 q 18 ( p q 3 ) 6 = ( p ) 6 ( q 3 ) 6 = p 1 ⋅ 6 q 3 ⋅ 6 = p 6 q 18
• ⓒ ( −1 t 2 ) 27 = ( −1 ) 27 ( t 2 ) 27 = −1 t 2 ⋅ 27 = −1 t 54 = − 1 t 54 ( −1 t 2 ) 27 = ( −1 ) 27 ( t 2 ) 27 = −1 t 2 ⋅ 27 = −1 t 54 = − 1 t 54
• ⓓ ( j 3 k −2 ) 4 = ( j 3 k 2 ) 4 = ( j 3 ) 4 ( k 2 ) 4 = j 3 ⋅ 4 k 2 ⋅ 4 = j 12 k 8 ( j 3 k −2 ) 4 = ( j 3 k 2 ) 4 = ( j 3 ) 4 ( k 2 ) 4 = j 3 ⋅ 4 k 2 ⋅ 4 = j 12 k 8
• ⓔ ( m −2 n −2 ) 3 = ( 1 m 2 n 2 ) 3 = ( 1 ) 3 ( m 2 n 2 ) 3 = 1 ( m 2 ) 3 ( n 2 ) 3 = 1 m 2 ⋅ 3 ⋅ n 2 ⋅ 3 = 1 m 6 n 6 ( m −2 n −2 ) 3 = ( 1 m 2 n 2 ) 3 = ( 1 ) 3 ( m 2 n 2 ) 3 = 1 ( m 2 ) 3 ( n 2 ) 3 = 1 m 2 ⋅ 3 ⋅ n 2 ⋅ 3 = 1 m 6 n 6
• ⓐ ( b 5 c ) 3 ( b 5 c ) 3
• ⓑ ( 5 u 8 ) 4 ( 5 u 8 ) 4
• ⓒ ( −1 w 3 ) 35 ( −1 w 3 ) 35
• ⓓ ( p −4 q 3 ) 8 ( p −4 q 3 ) 8
• ⓔ ( c −5 d −3 ) 4 ( c −5 d −3 ) 4

Simplifying Exponential Expressions

Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

Simplify each expression and write the answer with positive exponents only.

• ⓐ ( 6 m 2 n −1 ) 3 ( 6 m 2 n −1 ) 3
• ⓑ 17 5 ⋅ 17 −4 ⋅ 17 −3 17 5 ⋅ 17 −4 ⋅ 17 −3
• ⓒ ( u −1 v v −1 ) 2 ( u −1 v v −1 ) 2
• ⓓ ( −2 a 3 b −1 ) ( 5 a −2 b 2 ) ( −2 a 3 b −1 ) ( 5 a −2 b 2 )
• ⓔ ( x 2 2 ) 4 ( x 2 2 ) −4 ( x 2 2 ) 4 ( x 2 2 ) −4
• ⓕ ( 3 w 2 ) 5 ( 6 w −2 ) 2 ( 3 w 2 ) 5 ( 6 w −2 ) 2
• ⓐ ( 6 m 2 n −1 ) 3 = ( 6 ) 3 ( m 2 ) 3 ( n −1 ) 3 The power of a product rule = 6 3 m 2 ⋅ 3 n −1 ⋅ 3 The power rule = 216 m 6 n −3 Simplify . = 216 m 6 n 3 The negative exponent rule ( 6 m 2 n −1 ) 3 = ( 6 ) 3 ( m 2 ) 3 ( n −1 ) 3 The power of a product rule = 6 3 m 2 ⋅ 3 n −1 ⋅ 3 The power rule = 216 m 6 n −3 Simplify . = 216 m 6 n 3 The negative exponent rule
• ⓑ 17 5 ⋅ 17 −4 ⋅ 17 −3 = 17 5 − 4 − 3 The product rule = 17 −2 Simplify . = 1 17 2 or  1 289 The negative exponent rule 17 5 ⋅ 17 −4 ⋅ 17 −3 = 17 5 − 4 − 3 The product rule = 17 −2 Simplify . = 1 17 2 or  1 289 The negative exponent rule
• ⓒ ( u −1 v v −1 ) 2 = ( u −1 v ) 2 ( v −1 ) 2 The power of a quotient rule = u −2 v 2 v −2 The power of a product rule = u −2 v 2 − ( −2 ) The quotient rule = u −2 v 4 Simplify . = v 4 u 2 The negative exponent rule ( u −1 v v −1 ) 2 = ( u −1 v ) 2 ( v −1 ) 2 The power of a quotient rule = u −2 v 2 v −2 The power of a product rule = u −2 v 2 − ( −2 ) The quotient rule = u −2 v 4 Simplify . = v 4 u 2 The negative exponent rule
• ⓓ ( −2 a 3 b − 1 ) ( 5 a −2 b 2 ) = −2 ⋅ 5 ⋅ a 3 ⋅ a −2 ⋅ b −1 ⋅ b 2 Commutative and associative laws of multiplication = −10 ⋅ a 3 − 2 ⋅ b −1 + 2 The product rule = −10 a b Simplify . ( −2 a 3 b − 1 ) ( 5 a −2 b 2 ) = −2 ⋅ 5 ⋅ a 3 ⋅ a −2 ⋅ b −1 ⋅ b 2 Commutative and associative laws of multiplication = −10 ⋅ a 3 − 2 ⋅ b −1 + 2 The product rule = −10 a b Simplify .
• ⓔ ( x 2 2 ) 4 ( x 2 2 ) −4 = ( x 2 2 ) 4 − 4 The product rule = ( x 2 2 ) 0 Simplify . = 1 The zero exponent rule ( x 2 2 ) 4 ( x 2 2 ) −4 = ( x 2 2 ) 4 − 4 The product rule = ( x 2 2 ) 0 Simplify . = 1 The zero exponent rule
• ⓕ ( 3 w 2 ) 5 ( 6 w −2 ) 2 = ( 3 ) 5 ⋅ ( w 2 ) 5 ( 6 ) 2 ⋅ ( w −2 ) 2 The power of a product rule = 3 5 w 2 ⋅ 5 6 2 w −2 ⋅ 2 The power rule = 243 w 10 36 w −4 Simplify . = 27 w 10 − ( −4 ) 4 The quotient rule and reduce fraction = 27 w 14 4 Simplify . ( 3 w 2 ) 5 ( 6 w −2 ) 2 = ( 3 ) 5 ⋅ ( w 2 ) 5 ( 6 ) 2 ⋅ ( w −2 ) 2 The power of a product rule = 3 5 w 2 ⋅ 5 6 2 w −2 ⋅ 2 The power rule = 243 w 10 36 w −4 Simplify . = 27 w 10 − ( −4 ) 4 The quotient rule and reduce fraction = 27 w 14 4 Simplify .
• ⓐ ( 2 u v − 2 ) −3 ( 2 u v − 2 ) −3
• ⓑ x 8 ⋅ x −12 ⋅ x x 8 ⋅ x −12 ⋅ x
• ⓒ ( e 2 f − 3 f −1 ) 2 ( e 2 f − 3 f −1 ) 2
• ⓓ ( 9 r −5 s 3 ) ( 3 r 6 s −4 ) ( 9 r −5 s 3 ) ( 3 r 6 s −4 )
• ⓔ ( 4 9 t w −2 ) −3 ( 4 9 t w −2 ) 3 ( 4 9 t w −2 ) −3 ( 4 9 t w −2 ) 3
• ⓕ ( 2 h 2 k ) 4 ( 7 h −1 k 2 ) 2 ( 2 h 2 k ) 4 ( 7 h −1 k 2 ) 2

Using Scientific Notation

Recall at the beginning of the section that we found the number 1.3 × 10 13 1.3 × 10 13 when describing bits of information in digital images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?

A shorthand method of writing very small and very large numbers is called scientific notation , in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n . If you moved the decimal left as in a very large number, n n is positive. If you moved the decimal right as in a small large number, n n is negative.

For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

• Scientific Notation

A number is written in scientific notation if it is written in the form a × 10 n , a × 10 n , where 1 ≤ | a | < 10 1 ≤ | a | < 10 and n n is an integer.

Converting Standard Notation to Scientific Notation

Write each number in scientific notation.

ⓐ Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m

ⓑ Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m

ⓒ Number of stars in Andromeda Galaxy: 1,000,000,000,000

ⓓ Diameter of electron: 0.00000000000094 m

ⓔ Probability of being struck by lightning in any single year: 0.00000143

ⓐ 24,000,000,000,000,000,000,000 m 24,000,000,000,000,000,000,000 m ← 22 places 2.4 × 10 22 m 24,000,000,000,000,000,000,000 m 24,000,000,000,000,000,000,000 m ← 22 places 2.4 × 10 22 m

ⓑ 1,300,000,000,000,000,000,000 m 1,300,000,000,000,000,000,000 m ← 21 places 1.3 × 10 21 m 1,300,000,000,000,000,000,000 m 1,300,000,000,000,000,000,000 m ← 21 places 1.3 × 10 21 m

ⓒ 1,000,000,000,000 1,000,000,000,000 ← 12 places 1 × 10 12 1,000,000,000,000 1,000,000,000,000 ← 12 places 1 × 10 12

ⓓ 0.00000000000094 m 0.00000000000094 m → 13 places 9.4 × 10 −13 m 0.00000000000094 m 0.00000000000094 m → 13 places 9.4 × 10 −13 m

ⓔ 0.00000143 0.00000143 → 6 places 1.43 × 10 −6 0.00000143 0.00000143 → 6 places 1.43 × 10 −6

Observe that, if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive; and if the number is less than 1, as in examples d–e, the exponent is negative.

• ⓐ U.S. national debt per taxpayer (April 2014): $152,000
• ⓑ World population (April 2014): 7,158,000,000
• ⓒ World gross national income (April 2014): $85,500,000,000,000
• ⓓ Time for light to travel 1 m: 0.00000000334 s
• ⓔ Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal n n places to the right if n n is positive or n n places to the left if n n is negative and add zeros as needed. Remember, if n n is positive, the value of the number is greater than 1, and if n n is negative, the value of the number is less than one.

Converting Scientific Notation to Standard Notation

Convert each number in scientific notation to standard notation.

• ⓐ 3.547 × 10 14 3.547 × 10 14
• ⓑ −2 × 10 6 −2 × 10 6
• ⓒ 7.91 × 10 −7 7.91 × 10 −7
• ⓓ −8.05 × 10 −12 −8.05 × 10 −12
• ⓐ 3.547 × 10 14 3.54700000000000 → 14 places 354,700,000,000,000 3.547 × 10 14 3.54700000000000 → 14 places 354,700,000,000,000
• ⓑ −2 × 10 6 −2.000000 → 6 places −2,000,000 −2 × 10 6 −2.000000 → 6 places −2,000,000
• ⓒ 7.91 × 10 −7 0000007.91 → 7 places 0.000000791 7.91 × 10 −7 0000007.91 → 7 places 0.000000791
• ⓓ −8.05 × 10 −12 −000000000008.05 → 12 places −0.00000000000805 −8.05 × 10 −12 −000000000008.05 → 12 places −0.00000000000805
• ⓐ 7.03 × 10 5 7.03 × 10 5
• ⓑ −8.16 × 10 11 −8.16 × 10 11
• ⓒ −3.9 × 10 −13 −3.9 × 10 −13
• ⓓ 8 × 10 −6 8 × 10 −6

Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around 1.32 × 10 21 1.32 × 10 21 molecules of water and 1 L of water holds about 1.22 × 10 4 1.22 × 10 4 average drops. Therefore, there are approximately 3 ⋅ ( 1.32 × 10 21 ) ⋅ ( 1.22 × 10 4 ) ≈ 4.83 × 10 25 3 ⋅ ( 1.32 × 10 21 ) ⋅ ( 1.22 × 10 4 ) ≈ 4.83 × 10 25 atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product ( 7 × 10 4 ) ⋅ ( 5 × 10 6 ) = 35 × 10 10 . ( 7 × 10 4 ) ⋅ ( 5 × 10 6 ) = 35 × 10 10 . The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as 3.5 × 10. 3.5 × 10. That adds a ten to the exponent of the answer.

Perform the operations and write the answer in scientific notation.

• ⓐ ( 8.14 × 10 −7 ) ( 6.5 × 10 10 ) ( 8.14 × 10 −7 ) ( 6.5 × 10 10 )
• ⓑ ( 4 × 10 5 ) ÷ ( −1.52 × 10 9 ) ( 4 × 10 5 ) ÷ ( −1.52 × 10 9 )
• ⓒ ( 2.7 × 10 5 ) ( 6.04 × 10 13 ) ( 2.7 × 10 5 ) ( 6.04 × 10 13 )
• ⓓ ( 1.2 × 10 8 ) ÷ ( 9.6 × 10 5 ) ( 1.2 × 10 8 ) ÷ ( 9.6 × 10 5 )
• ⓔ ( 3.33 × 10 4 ) ( −1.05 × 10 7 ) ( 5.62 × 10 5 ) ( 3.33 × 10 4 ) ( −1.05 × 10 7 ) ( 5.62 × 10 5 )
• ⓐ ( 8.14 × 10 −7 ) ( 6.5 × 10 10 ) = ( 8.14 × 6.5 ) ( 10 −7 × 10 10 ) Commutative and associative properties of multiplication = ( 52.91 ) ( 10 3 ) Product rule of exponents = 5.291 × 10 4 Scientific notation ( 8.14 × 10 −7 ) ( 6.5 × 10 10 ) = ( 8.14 × 6.5 ) ( 10 −7 × 10 10 ) Commutative and associative properties of multiplication = ( 52.91 ) ( 10 3 ) Product rule of exponents = 5.291 × 10 4 Scientific notation
• ⓑ ( 4 × 10 5 ) ÷ ( −1.52 × 10 9 ) = ( 4 −1.52 ) ( 10 5 10 9 ) Commutative and associative properties of multiplication ≈ ( −2.63 ) ( 10 −4 ) Quotient rule of exponents = −2.63 × 10 −4 Scientific notation ( 4 × 10 5 ) ÷ ( −1.52 × 10 9 ) = ( 4 −1.52 ) ( 10 5 10 9 ) Commutative and associative properties of multiplication ≈ ( −2.63 ) ( 10 −4 ) Quotient rule of exponents = −2.63 × 10 −4 Scientific notation
• ⓒ ( 2.7 × 10 5 ) ( 6.04 × 10 13 ) = ( 2.7 × 6.04 ) ( 10 5 × 10 13 ) Commutative and associative properties of multiplication = ( 16.308 ) ( 10 18 ) Product rule of exponents = 1.6308 × 10 19 Scientific notation ( 2.7 × 10 5 ) ( 6.04 × 10 13 ) = ( 2.7 × 6.04 ) ( 10 5 × 10 13 ) Commutative and associative properties of multiplication = ( 16.308 ) ( 10 18 ) Product rule of exponents = 1.6308 × 10 19 Scientific notation
• ⓓ ( 1.2 × 10 8 ) ÷ ( 9.6 × 10 5 ) = ( 1.2 9.6 ) ( 10 8 10 5 ) Commutative and associative properties of multiplication = ( 0.125 ) ( 10 3 ) Quotient rule of exponents = 1.25 × 10 2 Scientific notation ( 1.2 × 10 8 ) ÷ ( 9.6 × 10 5 ) = ( 1.2 9.6 ) ( 10 8 10 5 ) Commutative and associative properties of multiplication = ( 0.125 ) ( 10 3 ) Quotient rule of exponents = 1.25 × 10 2 Scientific notation
• ⓔ ( 3.33 × 10 4 ) ( −1.05 × 10 7 ) ( 5.62 × 10 5 ) = [ 3.33 × ( −1.05 ) × 5.62 ] ( 10 4 × 10 7 × 10 5 ) ≈ ( −19.65 ) ( 10 16 ) = −1.965 × 10 17 ( 3.33 × 10 4 ) ( −1.05 × 10 7 ) ( 5.62 × 10 5 ) = [ 3.33 × ( −1.05 ) × 5.62 ] ( 10 4 × 10 7 × 10 5 ) ≈ ( −19.65 ) ( 10 16 ) = −1.965 × 10 17
• ⓐ ( −7.5 × 10 8 ) ( 1.13 × 10 −2 ) ( −7.5 × 10 8 ) ( 1.13 × 10 −2 )
• ⓑ ( 1.24 × 10 11 ) ÷ ( 1.55 × 10 18 ) ( 1.24 × 10 11 ) ÷ ( 1.55 × 10 18 )
• ⓒ ( 3.72 × 10 9 ) ( 8 × 10 3 ) ( 3.72 × 10 9 ) ( 8 × 10 3 )
• ⓓ ( 9.933 × 10 23 ) ÷ ( − 2.31 × 10 17 ) ( 9.933 × 10 23 ) ÷ ( − 2.31 × 10 17 )
• ⓔ ( −6.04 × 10 9 ) ( 7.3 × 10 2 ) ( −2.81 × 10 2 ) ( −6.04 × 10 9 ) ( 7.3 × 10 2 ) ( −2.81 × 10 2 )

Applying Scientific Notation to Solve Problems

In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.

The population was 308,000,000 = 3.08 × 10 8 . 308,000,000 = 3.08 × 10 8 .

The national debt was $ 17,547,000,000,000 ≈ $ 1.75 × 10 13 . $ 17,547,000,000,000 ≈ $ 1.75 × 10 13 .

To find the amount of debt per citizen, divide the national debt by the number of citizens.

The debt per citizen at the time was about $ 5.7 × 10 4 , $ 5.7 × 10 4 , or $57,000.

An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

Access these online resources for additional instruction and practice with exponents and scientific notation.

• Exponential Notation
• Properties of Exponents
• Zero Exponent
• Simplify Exponent Expressions
• Quotient Rule for Exponents
• Converting to Decimal Notation

1.2 Section Exercises

Is 2 3 2 3 the same as 3 2 ? 3 2 ? Explain.

When can you add two exponents?

What is the purpose of scientific notation?

Explain what a negative exponent does.

For the following exercises, simplify the given expression. Write answers with positive exponents.

15 −2 15 −2

3 2 × 3 3 3 2 × 3 3

4 4 ÷ 4 4 4 ÷ 4

( 2 2 ) −2 ( 2 2 ) −2

( 5 − 8 ) 0 ( 5 − 8 ) 0

11 3 ÷ 11 4 11 3 ÷ 11 4

6 5 × 6 −7 6 5 × 6 −7

( 8 0 ) 2 ( 8 0 ) 2

5 −2 ÷ 5 2 5 −2 ÷ 5 2

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

4 2 × 4 3 ÷ 4 −4 4 2 × 4 3 ÷ 4 −4

6 12 6 9 6 12 6 9

( 12 3 × 12 ) 10 ( 12 3 × 12 ) 10

10 6 ÷ ( 10 10 ) −2 10 6 ÷ ( 10 10 ) −2

7 −6 × 7 −3 7 −6 × 7 −3

( 3 3 ÷ 3 4 ) 5 ( 3 3 ÷ 3 4 ) 5

For the following exercises, express the decimal in scientific notation.

148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

1.6 × 10 10 1.6 × 10 10

9.8 × 10 −9 9.8 × 10 −9

a 3 a 2 a a 3 a 2 a

m n 2 m −2 m n 2 m −2

( b 3 c 4 ) 2 ( b 3 c 4 ) 2

( x −3 y 2 ) −5 ( x −3 y 2 ) −5

a b 2 ÷ d −3 a b 2 ÷ d −3

( w 0 x 5 ) −1 ( w 0 x 5 ) −1

m 4 n 0 m 4 n 0

y −4 ( y 2 ) 2 y −4 ( y 2 ) 2

p −4 q 2 p 2 q −3 p −4 q 2 p 2 q −3

( l × w ) 2 ( l × w ) 2

( y 7 ) 3 ÷ x 14 ( y 7 ) 3 ÷ x 14

( a 2 3 ) 2 ( a 2 3 ) 2

( 25 m ) ÷ ( 5 0 m ) ( 25 m ) ÷ ( 5 0 m )

( 16 x ) 2 y −1 ( 16 x ) 2 y −1

2 3 ( 3 a ) −2 2 3 ( 3 a ) −2

( m a 6 ) 2 1 m 3 a 2 ( m a 6 ) 2 1 m 3 a 2

( b −3 c ) 3 ( b −3 c ) 3

( x 2 y 13 ÷ y 0 ) 2 ( x 2 y 13 ÷ y 0 ) 2

( 9 z 3 ) −2 y ( 9 z 3 ) −2 y

Real-World Applications

To reach escape velocity, a rocket must travel at the rate of 2.2 × 10 6 2.2 × 10 6 ft/min. Rewrite the rate in standard notation.

A dime is the thinnest coin in U.S. currency. A dime’s thickness measures 1.35 × 10 −3 1.35 × 10 −3 m. Rewrite the number in standard notation.

The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.

A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was $ 1.71496 × 10 13 . $ 1.71496 × 10 13 . Rewrite the GDP in standard notation.

One picometer is approximately 3.397 × 10 −11 3.397 × 10 −11 in. Rewrite this length using standard notation.

The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

( 12 3 m 33 4 −3 ) 2 ( 12 3 m 33 4 −3 ) 2

17 3 ÷ 15 2 x 3 17 3 ÷ 15 2 x 3

( 3 2 a 3 ) −2 ( a 4 2 2 ) 2 ( 3 2 a 3 ) −2 ( a 4 2 2 ) 2

( 6 2 −24 ) 2 ÷ ( x y ) −5 ( 6 2 −24 ) 2 ÷ ( x y ) −5

m 2 n 3 a 2 c −3 ⋅ a −7 n −2 m 2 c 4 m 2 n 3 a 2 c −3 ⋅ a −7 n −2 m 2 c 4

( x 6 y 3 x 3 y −3 ⋅ y −7 x −3 ) 10 ( x 6 y 3 x 3 y −3 ⋅ y −7 x −3 ) 10

( ( a b 2 c ) −3 b −3 ) 2 ( ( a b 2 c ) −3 b −3 ) 2

Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is 6.0221413 × 10 23 . 6.0221413 × 10 23 . Write Avogadro’s constant in standard notation.

Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as 6.62606957 × 10 −34 . 6.62606957 × 10 −34 . Write Planck’s constant in standard notation.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra
• Publication date: Feb 13, 2015
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra/pages/1-2-exponents-and-scientific-notation

© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• Solutions Integral Calculator Derivative Calculator Algebra Calculator Matrix Calculator More...
• Graphing Line Graph Exponential Graph Quadratic Graph Sine Graph More...
• Calculators BMI Calculator Compound Interest Calculator Percentage Calculator Acceleration Calculator More...
• Geometry Pythagorean Theorem Calculator Circle Area Calculator Isosceles Triangle Calculator Triangles Calculator More...
• Tools Notebook Groups Cheat Sheets Worksheets Study Guides Practice Verify Solution

Study Guides > College Algebra CoRequisite Course

Scientific notation, learning outcomes.

• Convert standard notation to scientific notation.
• Convert from scientific to standard notation.
• Apply scientific notation in an application.

A General Note: Scientific Notation

Example: converting standard notation to scientific notation.

• Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
• Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
• Number of stars in Andromeda Galaxy: 1,000,000,000,000
• Diameter of electron: 0.00000000000094 m
• Probability of being struck by lightning in any single year: 0.00000143

1. [latex]\begin{align}&\underset{\leftarrow 22\text{ places}}{{24,000,000,000,000,000,000,000\text{ m}}} \\ &2.4\times {10}^{22}\text{ m} \\ \text{ } \end{align}[/latex]

2. [latex]\begin{align}&\underset{\leftarrow 21\text{ places}}{{1,300,000,000,000,000,000,000\text{ m}}} \\ &1.3\times {10}^{21}\text{ m} \\ &\text{ } \end{align}[/latex]

3. [latex]\begin{align}&\underset{\leftarrow 12\text{ places}}{{1,000,000,000,000}} \\ &1\times {10}^{12} \\ \text{ }\end{align}[/latex]

4. [latex]\begin{align}&\underset{\rightarrow 6\text{ places}}{{0.00000000000094\text{ m}}} \\ &9.4\times {10}^{-13}\text{ m} \\ \text{ }\end{align}[/latex]

5. [latex]\begin{align}\underset{\to 6\text{ places}}{{0.00000143}} \\ 1.43\times {10}^{-6} \\ \text{ }\end{align}[/latex]

Analysis of the Solution

• U.S. national debt per taxpayer (April 2014): $152,000
• World population (April 2014): 7,158,000,000
• World gross national income (April 2014): $85,500,000,000,000
• Time for light to travel 1 m: 0.00000000334 s
• Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
• [latex]$1.52\times {10}^{5}[/latex]
• [latex]7.158\times {10}^{9}[/latex]
• [latex]$8.55\times {10}^{13}[/latex]
• [latex]3.34\times {10}^{-9}[/latex]
• [latex]7.15\times {10}^{-8}[/latex]

Converting from Scientific to Standard Notation

Example: converting scientific notation to standard notation.

• [latex]3.547\times {10}^{14}[/latex]
• [latex]-2\times {10}^{6}[/latex]
• [latex]7.91\times {10}^{-7}[/latex]
• [latex]-8.05\times {10}^{-12}[/latex]

Answer: 1. [latex-display]\begin{align}&3.547\times {10}^{14} \\ &\underset{\to 14\text{ places}}{{3.54700000000000}} \\ &354,700,000,000,000 \\ \text{ }\end{align}[/latex-display] 2. [latex-display]\begin{align}&-2\times {10}^{6} \\ &\underset{\to 6\text{ places}}{{-2.000000}} \\ &-2,000,000 \\ \text{ }\end{align}[/latex-display] 3. [latex-display]\begin{align}&7.91\times {10}^{-7} \\ &\underset{\to 7\text{ places}}{{0000007.91}} \\ &0.000000791 \\ \text{ }\end{align}[/latex-display] 4. [latex-display]\begin{align}&-8.05\times {10}^{-12} \\ &\underset{\to 12\text{ places}}{{-000000000008.05}} \\ &-0.00000000000805 \\ \text{ }\end{align}[/latex-display]

• [latex]7.03\times {10}^{5}[/latex]
• [latex]-8.16\times {10}^{11}[/latex]
• [latex]-3.9\times {10}^{-13}[/latex]
• [latex]8\times {10}^{-6}[/latex]
• [latex]703,000[/latex]
• [latex]-816,000,000,000[/latex]
• [latex]-0.00000000000039[/latex]
• [latex]0.000008[/latex]

Using Scientific Notation in Applications

What properties of numbers enable operations on scientific notation, example: using scientific notation.

• [latex]\left(8.14\times {10}^{-7}\right)\left(6.5\times {10}^{10}\right)[/latex]
• [latex]\left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)[/latex]
• [latex]\left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)[/latex]
• [latex]\left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)[/latex]
• [latex]\left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)[/latex]

Answer: 1. [latex-display]\begin{align}\left(8.14 \times 10^{-7}\right)\left(6.5 \times 10^{10}\right) & =\left(8.14 \times 6.5\right)\left(10^{-7} \times 10^{10}\right) && \text{Commutative and associative properties of multiplication} \\ & =\left(52.91\right)\left(10^{3}\right) && \text{Product rule of exponents} \\ & =5.291 \times 10^{4} && \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 2. [latex-display]\begin{align} \left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)& = \left(\frac{4}{-1.52}\right)\left(\frac{{10}^{5}}{{10}^{9}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(-2.63\right)\left({10}^{-4}\right)&& \text{Quotient rule of exponents} \\ & = -2.63\times {10}^{-4}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 3. [latex-display]\begin{align} \left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)& = \left(2.7\times 6.04\right)\left({10}^{5}\times {10}^{13}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(16.308\right)\left({10}^{18}\right)&& \text{Product rule of exponents} \\ & = 1.6308\times {10}^{19}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 4. [latex-display]\begin{align} \left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)& = \left(\frac{1.2}{9.6}\right)\left(\frac{{10}^{8}}{{10}^{5}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(0.125\right)\left({10}^{3}\right)&& \text{Quotient rule of exponents} \\ & = 1.25\times {10}^{2}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex-display] 5. [latex-display]\begin{align} \left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)& = \left[3.33\times \left(-1.05\right)\times 5.62\right]\left({10}^{4}\times {10}^{7}\times {10}^{5}\right) \\ & \approx \left(-19.65\right)\left({10}^{16}\right) \\ & = -1.965\times {10}^{17} \end{align}[/latex-display]

• [latex]\left(-7.5\times {10}^{8}\right)\left(1.13\times {10}^{-2}\right)[/latex]
• [latex]\left(1.24\times {10}^{11}\right)\div \left(1.55\times {10}^{18}\right)[/latex]
• [latex]\left(3.72\times {10}^{9}\right)\left(8\times {10}^{3}\right)[/latex]
• [latex]\left(9.933\times {10}^{23}\right)\div \left(-2.31\times {10}^{17}\right)[/latex]
• [latex]\left(-6.04\times {10}^{9}\right)\left(7.3\times {10}^{2}\right)\left(-2.81\times {10}^{2}\right)[/latex]
• [latex]-8.475\times {10}^{6}[/latex]
• [latex]8\times {10}^{-8}[/latex]
• [latex]2.976\times {10}^{13}[/latex]
• [latex]-4.3\times {10}^{6}[/latex]
• [latex]\approx 1.24\times {10}^{15}[/latex]

Example: Applying Scientific Notation to Solve Problems

Answer: The population was [latex]308,000,000=3.08\times {10}^{8}[/latex]. The national debt was [latex]\$ 17,547,000,000,000 \approx \$1.75 \times 10^{13}[/latex]. To find the amount of debt per citizen, divide the national debt by the number of citizens. [latex]\begin{align} \left(1.75\times {10}^{13}\right)\div \left(3.08\times {10}^{8}\right)& = \left(\frac{1.75}{3.08}\right)\cdot \left(\frac{{10}^{13}}{{10}^{8}}\right) \\ & \approx 0.57\times {10}^{5}\hfill \\ & = 5.7\times {10}^{4} \end{align}[/latex] The debt per citizen at the time was about [latex]\$5.7\times {10}^{4}[/latex], or $57,000.

Answer: Number of cells: [latex]3\times {10}^{13}[/latex]; length of a cell: [latex]8\times {10}^{-6}[/latex] m; total length: [latex]2.4\times {10}^{8}[/latex] m or [latex]240,000,000[/latex] m.

Licenses & Attributions

Cc licensed content, original.

• Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution .

CC licensed content, Shared previously

• College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/ [email protected] .
• Examples: Write a Number in Scientific Notation. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution .
• Question ID 2466. Authored by: Bryan Jones. License: CC BY: Attribution . License terms: IMathAS Community License, CC-BY + GPL.
• Question ID 2836, 2839, 2841, 2824, 2828, 2830. Authored by: Gregg Harbaugh. License: CC BY: Attribution . License terms: IMathAS Community License, CC-BY + GPL.
• Question ID 2874. Authored by: David Lippman. License: CC BY: Attribution . License terms: IMathAS Community License, CC-BY + GPL.
• Question ID 3295. Authored by: Norm Friehauf. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.
• Question ID 101856, 102452. Authored by: Roy Shahbazian. License: CC BY: Attribution . License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Specific attribution

• College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/ [email protected] :1/Preface. License: CC BY: Attribution .

We want your feedback

Please add a message.

Message received. Thanks for the feedback.

Scientific Notation and Standard Form

Formula, practice problems

Scientific notation is just a short hand way of expressing gigantic numbers like 1,300,000 or incredibly small numbers like 0.0000000000045. Also known as exponential form, scientific notation has been one of the oldest mathematical approaches. It is favored by many practicioners. If numbers are too big or too small to be simply calculated, people reffer to scientific notation to handle these circumstances. This method is used by engineers, mathematicians, scientists.

An example of scientific notation is 1.3 ×10 6 which is just a different way of expressing the standard notation of the number 1,300,000. Standard notation is the normal way of writing numbers.

Key Vocabulary mantissa = this is the integer or first digit in any Scientific Notation. For example in 1.3 ×10 6 , the mantissa is the "1"

Other examples:

• Scientific Notation Worksheets (free pdfs with answer keys on scientific notation)
• scientific notation in action 
• Problem Set #2

General Formula of Scientific Notation

The general from of a number in scientific notation is:

a ×10 n where  1 ≤ a ≤ 10 and n is an integer .  In other words the number  that we'll call "a"  is is multiplied by 10, raised to some exponent n. This number "a" must be no smaller than 1 and no larger than 10. To illustrate this definition examine the following: 1.4 ×10 4 is a proper example of scientific notation because

• 1.4, which is "a" in this example, is not smaller than 1 and not larger than 10 so it's ok.
• 10's exponent is the integer 4.
• .9 which is "a" in this example, is smaller than 1 which is not allowed in scientific notation
• 10's exponent is not an integer .
• 4.34, which is "a" in this example, is not smaller than 1 and not larger than 10
• 10's exponent is the integer -55. Integers can be negative

Practice Problems

In the following sentences, convert from scientific notation to standard form.

In the following sentences, convert from standard form to scientific notation.

In the following sentences, state whether or not the given number is in Scientific Notation and explain your answer. 

Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com and around the web

Have an account?

Suggestions for you See more

Scientific Notation

18.6k plays, multiplication standard algorithm, 3rd -  4th  , expanded notation, multiplying and dividing scientific nota..., converting scientific notation, writing in standard form.

Multiplying & Dividing in Scientific Not...

Mathematics.

Multiplying & Dividing in Scientific Notation Practice

24 questions

Introducing new   Paper mode

No student devices needed.   Know more

• 1. Multiple Choice Edit 45 seconds 1 pt When dividing numbers in scientific notation, what do you do to the exponents? For example: (4×10⁵) ∕ (3×10²) Add them Subtract them Keep them the same Divide them
• 2. Multiple Choice Edit 45 seconds 1 pt In what types of problems do you have to make sure the exponents are the same before beginning to compute? Adding and Subtracting Multiplying and Dividing Subtracting and Dividing Adding and Multiplying
• 3. Multiple Choice Edit 45 seconds 1 pt When multiplying numbers in scientific notation, what do you do to the leading numbers? For example: (4×10⁵)(3×10⁻²) Keep them the same Divide them Add them Multiply them
• 4. Multiple Choice Edit 2 minutes 1 pt Evaluate. Leave your answer in scientific notation.  (3.4×10⁴)(2×10⁴) 6.8×10⁴ 5.4×10⁴ 6.8×10⁸ 5.4×10⁸
• 5. Multiple Choice Edit 2 minutes 1 pt Evaluate. Leave your answer in scientific notation.  (9.6×10⁸)  ⁄  (3×10⁴) (hint - it's division...) 6.6×10² 3.2×10¹² 3.2×10⁴ 6.6×10⁴
• 6. Multiple Choice Edit 2 minutes 1 pt Order from least to greatest... a) 4.5×10⁻³ b) 0.000045 c) 3.1×10⁻³   d) -45           D , B , C,  A B , C , D , A C , A , D , B A , C , D , B
• 7. Multiple Choice Edit 1 minute 1 pt Which inequality sign should be placed between these numbers to make it a true/correct statement? 7.6×10⁴      ?       0.00076 < > =
• 8. Multiple Choice Edit 1 minute 1 pt Determine the missing value. (5.4×10 ? )(3.2×10³) = 17.28×10 15 12 8 18 5
• 10. Multiple Choice Edit 2 minutes 1 pt Multiply: (9.4 x 10 6 )(3.2 x 10 5 ) 30.08 x 10 11 3.8 x 10 1 3.008 x 10 12 2.9375 x 10 1
• 11. Multiple Choice Edit 3 minutes 1 pt (9.6×10 3 ) × (6.7×10 2 ) 64.32×10 5 6.432×10 6 64.32×10 6 64.32×10 5
• 12. Multiple Choice Edit 2 minutes 1 pt Solve: (6 x 10 6 ) / (2 x 10 3 ) 12 x 10 3 3 x 10 9 1.2 x 10 4 3 x 10 3
• 13. Multiple Choice Edit 5 minutes 1 pt (2 x 10 9 )(4 x 10 -4 ) 8 x 10 13 8 x 10 36 8 x 10 5 8 x 10 -36
• 14. Multiple Choice Edit 5 minutes 1 pt (5 x 10 6 )(5 x 10 7 ) 2.5 x 10 12 2.5 x 10 14 25 x 10 12 25 x 10 13

(20 x 10 50 ) / (4 x 10 15 )

2.4 x 10 51

Write 7.113 x 10 7 in standard form.

0.0000007113

• 17. Multiple Choice Edit 45 seconds 1 pt Write 0.00000707 in scientific notation. 7.07 x 10 -6 7.07 x 10 6 707 x 10 8 707 x 10 -8
• 18. Multiple Choice Edit 30 seconds 1 pt Write 7.8 x 10 -3   in standard form. 0.0078 7,800 78 0.078
• 19. Multiple Choice Edit 30 seconds 1 pt Write 2.08 x 10 2   in standard form. 208 0.0208 20,800 0.208
• 20. Multiple Choice Edit 30 seconds 1 pt UNDERSTANDING If the exponent is a negative number... you will get a large number you will get a small number
• 21. Multiple Choice Edit 30 seconds 1 pt UNDERSTANDING: The first number in scientific notation must be... between and including 0 and 10 between and including 1 and 9.9999 between and including 0 and 9.999 between and including 1 and 10 
• 22. Multiple Choice Edit 45 seconds 1 pt Which of the following is correct scientific notation? 20.35 x 10 4 .2035 x 10 4 2035 4 2.035 x10 4
• 23. Multiple Choice Edit 30 seconds 1 pt Which is the smallest? 1.3 x 10 20 2.9 x 10 21 9.5 x 10 32 8.4 x 10 19 1.3 x 10 20 2.9 x 10 21 9.5 x 10 32 8.4 x 10 19
• 24. Multiple Choice Edit 2 minutes 1 pt ANY number raised to a power of 0 will be.... 10 itself 1 0

Explore all questions with a free account

Continue with email

Continue with phone

• Add to Home
• TigerMilk.Education GmbH
• Privacy policy
• Terms of service

Copyright Ⓒ 2013-2024 tiger-algebra.com

• Arabic | اللغة العربية
• Spanish | Español
• French | Français
• German | Deutsch
• Hebrew | עִברִית
• Italian | Italiano
• Indonesian | Bahasa Indonesia
• Portuguese | Português
• Russian | Русский
• Serbian | Srpski
• Vietnamese | Tiếng Việt
• Chinese | 中文
• Hindi | हिंदी
• Bengali | বাংলা
• Filipino | Filipino
• Korean | 한국인
• Marathi | मराठी
• Tamil | தமிழ்
• Telugu | తెలుగు
• Turkish | Türkçe
• Japanese | 日本語

This site is best viewed with Javascript. If you are unable to turn on Javascript, please click here .

Solution - Scientific notation/Standard form

Other Ways to Solve

• Operations with Numbers in Scientific Notation
• Order of operations
• Alternative approach

Step-by-step explanation

1. convert to decimal notation.

1.13 ⋅ 10 13

The exponent is 13, making it 10 to the power of 13. As the exponent is positive, the solution is a number greater than the origin or base number. To find our answer, we move the decimal to the right 13 time(s):

1.13 -> 11300000000000

2. Final result

11300000000000

How did we do?

Why learn this

Scientific notation, or standard form, makes things easier when working with very small or very big numbers, both of which come up frequently in the fields of science and engineering. It is used in science, for example, to convey the mass of the heavenly bodies: Jupiter’s mass is 1.898 ⋅ 10 27 kg, which is easier to comprehend than writing the number 1,898 followed by 24 zeroes. Scientific notation also makes solving problems that use such high or low numbers more straightforward.

Terms and topics

• Scientific notation

Related links

• How to Write in Scientific Notation | Dummies
• Concept of Scientific Notation | Brightstorm
• Introduction to scientific notation | Khan Academy

Latest Related Drills Solved

• 99110121132.143
• 60160260360
• 100150200250
• 175150125100
• 99148197246

Module 1: Algebra Essentials

Scientific notation, learning outcomes.

• Convert standard notation to scientific notation.
• Convert from scientific to standard notation.
• Apply scientific notation in an application.

A shorthand method of writing very small and very large numbers is called scientific notation , in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n . If you moved the decimal left as in a very large number, [latex]n[/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[/latex] is negative.

For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

A General Note: Scientific Notation

A number is written in scientific notation if it is written in the form [latex]a\times {10}^{n}[/latex], where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer.

Example: Converting Standard Notation to Scientific Notation

Write each number in scientific notation.

• Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
• Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
• Number of stars in Andromeda Galaxy: 1,000,000,000,000
• Diameter of electron: 0.00000000000094 m
• Probability of being struck by lightning in any single year: 0.00000143

1. [latex]\begin{align}&\underset{\leftarrow 22\text{ places}}{{24,000,000,000,000,000,000,000\text{ m}}} \\ &2.4\times {10}^{22}\text{ m} \\ \text{ } \end{align}[/latex]

2. [latex]\begin{align}&\underset{\leftarrow 21\text{ places}}{{1,300,000,000,000,000,000,000\text{ m}}} \\ &1.3\times {10}^{21}\text{ m} \\ &\text{ } \end{align}[/latex]

3. [latex]\begin{align}&\underset{\leftarrow 12\text{ places}}{{1,000,000,000,000}} \\ &1\times {10}^{12} \\ \text{ }\end{align}[/latex]

4. [latex]\begin{align}&\underset{\rightarrow 6\text{ places}}{{0.00000000000094\text{ m}}} \\ &9.4\times {10}^{-13}\text{ m} \\ \text{ }\end{align}[/latex]

5. [latex]\begin{align}\underset{\to 6\text{ places}}{{0.00000143}} \\ 1.43\times {10}^{-6} \\ \text{ }\end{align}[/latex]

Analysis of the Solution

Observe that, if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive; and if the number is less than 1, as in examples d–e, the exponent is negative.

• U.S. national debt per taxpayer (April 2014): $152,000
• World population (April 2014): 7,158,000,000
• World gross national income (April 2014): $85,500,000,000,000
• Time for light to travel 1 m: 0.00000000334 s
• Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
• [latex]$1.52\times {10}^{5}[/latex]
• [latex]7.158\times {10}^{9}[/latex]
• [latex]$8.55\times {10}^{13}[/latex]
• [latex]3.34\times {10}^{-9}[/latex]
• [latex]7.15\times {10}^{-8}[/latex]

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal [latex]n[/latex] places to the right if [latex]n[/latex] is positive or [latex]n[/latex] places to the left if [latex]n[/latex] is negative and add zeros as needed. Remember, if [latex]n[/latex] is positive, the value of the number is greater than 1, and if [latex]n[/latex] is negative, the value of the number is less than one.

Example: Converting Scientific Notation to Standard Notation

Convert each number in scientific notation to standard notation.

• [latex]3.547\times {10}^{14}[/latex]
• [latex]-2\times {10}^{6}[/latex]
• [latex]7.91\times {10}^{-7}[/latex]
• [latex]-8.05\times {10}^{-12}[/latex]

1. [latex]\begin{align}&3.547\times {10}^{14} \\ &\underset{\to 14\text{ places}}{{3.54700000000000}} \\ &354,700,000,000,000 \\ \text{ }\end{align}[/latex]

2. [latex]\begin{align}&-2\times {10}^{6} \\ &\underset{\to 6\text{ places}}{{-2.000000}} \\ &-2,000,000 \\ \text{ }\end{align}[/latex]

3. [latex]\begin{align}&7.91\times {10}^{-7} \\ &\underset{\to 7\text{ places}}{{0000007.91}} \\ &0.000000791 \\ \text{ }\end{align}[/latex]

4. [latex]\begin{align}&-8.05\times {10}^{-12} \\ &\underset{\to 12\text{ places}}{{-000000000008.05}} \\ &-0.00000000000805 \\ \text{ }\end{align}[/latex]

• [latex]7.03\times {10}^{5}[/latex]
• [latex]-8.16\times {10}^{11}[/latex]
• [latex]-3.9\times {10}^{-13}[/latex]
• [latex]8\times {10}^{-6}[/latex]
• [latex]703,000[/latex]
• [latex]-816,000,000,000[/latex]
• [latex]-0.00000000000039[/latex]
• [latex]0.000008[/latex]

Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around [latex]1.32\times {10}^{21}[/latex] molecules of water and 1 L of water holds about [latex]1.22\times {10}^{4}[/latex] average drops. Therefore, there are approximately [latex]3\cdot \left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)\approx 4.83\times {10}^{25}[/latex] atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For example, consider the product [latex]\left(7\times {10}^{4}\right)\cdot \left(5\times {10}^{6}\right)=35\times {10}^{10}[/latex]. The answer is not in proper scientific notation because 35 is greater than 10. Consider 35 as [latex]3.5\times 10[/latex]. That adds a ten to the exponent of the answer.

Example: Using Scientific Notation

Perform the operations and write the answer in scientific notation.

• [latex]\left(8.14\times {10}^{-7}\right)\left(6.5\times {10}^{10}\right)[/latex]
• [latex]\left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)[/latex]
• [latex]\left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)[/latex]
• [latex]\left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)[/latex]
• [latex]\left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)[/latex]

1. [latex]\begin{align}\left(8.14 \times 10^{-7}\right)\left(6.5 \times 10^{10}\right) & =\left(8.14 \times 6.5\right)\left(10^{-7} \times 10^{10}\right) && \text{Commutative and associative properties of multiplication} \\ & =\left(52.91\right)\left(10^{3}\right) && \text{Product rule of exponents} \\ & =5.291 \times 10^{4} && \text{Scientific notation} \\ \text{ } \end{align}[/latex]

2. [latex]\begin{align} \left(4\times {10}^{5}\right)\div \left(-1.52\times {10}^{9}\right)& = \left(\frac{4}{-1.52}\right)\left(\frac{{10}^{5}}{{10}^{9}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(-2.63\right)\left({10}^{-4}\right)&& \text{Quotient rule of exponents} \\ & = -2.63\times {10}^{-4}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex]

3. [latex]\begin{align} \left(2.7\times {10}^{5}\right)\left(6.04\times {10}^{13}\right)& = \left(2.7\times 6.04\right)\left({10}^{5}\times {10}^{13}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(16.308\right)\left({10}^{18}\right)&& \text{Product rule of exponents} \\ & = 1.6308\times {10}^{19}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex]

4. [latex]\begin{align} \left(1.2\times {10}^{8}\right)\div \left(9.6\times {10}^{5}\right)& = \left(\frac{1.2}{9.6}\right)\left(\frac{{10}^{8}}{{10}^{5}}\right)&& \text{Commutative and associative properties of multiplication} \\ & = \left(0.125\right)\left({10}^{3}\right)&& \text{Quotient rule of exponents} \\ & = 1.25\times {10}^{2}&& \text{Scientific notation} \\ \text{ } \end{align}[/latex]

5. [latex]\begin{align} \left(3.33\times {10}^{4}\right)\left(-1.05\times {10}^{7}\right)\left(5.62\times {10}^{5}\right)& = \left[3.33\times \left(-1.05\right)\times 5.62\right]\left({10}^{4}\times {10}^{7}\times {10}^{5}\right) \\ & \approx \left(-19.65\right)\left({10}^{16}\right) \\ & = -1.965\times {10}^{17} \end{align}[/latex]

• [latex]\left(-7.5\times {10}^{8}\right)\left(1.13\times {10}^{-2}\right)[/latex]
• [latex]\left(1.24\times {10}^{11}\right)\div \left(1.55\times {10}^{18}\right)[/latex]
• [latex]\left(3.72\times {10}^{9}\right)\left(8\times {10}^{3}\right)[/latex]
• [latex]\left(9.933\times {10}^{23}\right)\div \left(-2.31\times {10}^{17}\right)[/latex]
• [latex]\left(-6.04\times {10}^{9}\right)\left(7.3\times {10}^{2}\right)\left(-2.81\times {10}^{2}\right)[/latex]
• [latex]-8.475\times {10}^{6}[/latex]
• [latex]8\times {10}^{-8}[/latex]
• [latex]2.976\times {10}^{13}[/latex]
• [latex]-4.3\times {10}^{6}[/latex]
• [latex]\approx 1.24\times {10}^{15}[/latex]

Example: Applying Scientific Notation to Solve Problems

In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.

The national debt was [latex]\$ 17,547,000,000,000 \approx \$1.75 \times 10^{13}[/latex].

To find the amount of debt per citizen, divide the national debt by the number of citizens.

The debt per citizen at the time was about [latex]\$5.7\times {10}^{4}[/latex], or $57,000.

An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

Number of cells: [latex]3\times {10}^{13}[/latex]; length of a cell: [latex]8\times {10}^{-6}[/latex] m; total length: [latex]2.4\times {10}^{8}[/latex] m or [latex]240,000,000[/latex] m.

Contribute!

Improve this page Learn More

• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
• College Algebra. Authored by : Abramson, Jay et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Examples: Write a Number in Scientific Notation. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/fsNu3AdIgdk . License : CC BY: Attribution
• Question ID 2466. Authored by : Bryan Jones. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Question ID 2836, 2839, 2841, 2824, 2828, 2830. Authored by : Gregg Harbaugh. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Question ID 2874. Authored by : David Lippman. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Question ID 3295. Authored by : Norm Friehauf. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• Question ID 101856, 102452. Authored by : Roy Shahbazian. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
• College Algebra. Authored by : OpenStax College Algebra. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution

Scientific Notation Calculator

Enter a regular number below which you want to convert to scientific notation.

The scientific notation calculator converts the given regular number to scientific notation.

A regular number is converted to scientific notation by moving the decimal point such that there will be only one non-zero digit to the left of the decimal point. The number of decimal places you move will be the exponent of 10.

If you move the decimal point to the left, the exponent of 10 will be positive. If you move decimal point to the right, the exponent of 10 will be negative.

Click the blue arrow to submit. Choose "Convert to Scientific Notation" from the topic selector and click to see the result in our Pre-Algebra Calculator!

Convert to Scientific Notation

Popular Problems

Convert to Scientific Notation 369000 Convert to Scientific Notation 741 Convert to Scientific Notation 0.0000725 Convert to Scientific Notation 0.0006 Convert to Scientific Notation 123

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&