Contents

Integer Exponents and Scientific Notation

Learning Objectives

By the end of this section, you will be able to:

Use the definition of a negative exponent
Simplify expressions with integer exponents
Convert from decimal notation to scientific notation
Convert scientific notation to decimal form
Multiply and divide using scientific notation

Be Prepared 10.5

Before you get started, take this readiness quiz.

What is the place value of the $66$ in the number $64,891 ? 64,891 ?$
If you missed this problem, review Example 1.3.
Name the decimal $0.0012 . 0.0012 .$
If you missed this problem, review [link].
Subtract: $5 - (−3) . 5 - (−3) .$
If you missed this problem, review Example 3.37.

Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.

Quotient Property of Exponents

If $a a$ is a real number, $a \neq 0, a \neq 0,$ and $m, n m,$ are whole numbers, then

\frac{a^{m}}{a^{n}} = a^{m - n}, m > n and \frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}, n > m \frac{a^{m}}{a^{n}} = a^{m - n},

What if we just subtract exponents, regardless of which is larger? Let’s consider $\frac{x^{2}}{x^{5}} . \frac{x^{2}}{x^{5}} .$

We subtract the exponent in the denominator from the exponent in the numerator.

\frac{x^{2}}{x^{5}} \frac{x^{2}}{x^{5}}

x^{2 - 5} x^{2 - 5}

x^{−3} x^{−3}

We can also simplify $\frac{x^{2}}{x^{5}} \frac{x^{2}}{x^{5}}$ by dividing out common factors: $\frac{x^{2}}{x^{5}} . \frac{x^{2}}{x^{5}} .$

$A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.$

This implies that $x^{−3} = \frac{1}{x^{3}} x^{−3} = \frac{1}{x^{3}}$ and it leads us to the definition of a negative exponent.

Negative Exponent

If $n n$ is a positive integer and $a \neq 0, a \neq 0,$ then $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$

The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

Example 10.63

Simplify:

ⓐ $4^{−2}$
ⓑ $10^{−3}$

Show solution

Solution

ⓐ
	$4^{−2} 4^{−2}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{4^{2}} \frac{1}{4^{2}}$
Simplify.	$\frac{1}{16} \frac{1}{16}$

ⓑ
	$10^{−3} 10^{−3}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{10^{3}} \frac{1}{10^{3}}$
Simplify.	$\frac{1}{1000} \frac{1}{1000}$

Try It 10.125

Simplify:

ⓐ $2^{−3}$
ⓑ $10^{−2}$

Try It 10.126

Simplify:

ⓐ $3^{−2}$
ⓑ $10^{−4}$

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

Example 10.64

Simplify:

ⓐ ${(−3)}^{−2}$
ⓑ ${−3}^{−2}$

Show solution

Solution

The negative in the exponent does not affect the sign of the base.

ⓐ
The exponent applies to the base, $- 3 - 3$ .	${(−3)}^{−2} {(−3)}^{−2}$
Take the reciprocal of the base and change the sign of the exponent.	$\frac{1}{{(−3)}^{2}} \frac{1}{{(−3)}^{2}}$
Simplify.	$\frac{1}{9} \frac{1}{9}$

ⓑ
The expression $- 3^{−2} - 3^{−2}$ means "find the opposite of $3^{−2} 3^{−2}$ ". The exponent applies only to the base, 3.	$- 3^{−2} - 3^{−2}$
Rewrite as a product with −1.	$−1 \cdot 3^{−2} −1 \cdot 3^{−2}$
Take the reciprocal of the base and change the sign of the exponent.	$−1 \cdot \frac{1}{3^{2}} −1 \cdot \frac{1}{3^{2}}$
Simplify.	$- \frac{1}{9} - \frac{1}{9}$

Try It 10.127

Simplify:

ⓐ ${(−5)}^{−2}$
ⓑ $- 5^{−2}$

Try It 10.128

Simplify:

ⓐ ${(−2)}^{−2}$
ⓑ ${−2}^{−2}$

We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.

Example 10.65

Simplify:

ⓐ $4 \cdot 2^{−1}$
ⓑ ${(4 \cdot 2)}^{−1}$

Show solution

Solution

Remember to always follow the order of operations.

ⓐ
Do exponents before multiplication.	$4 \cdot 2^{−1} 4 \cdot 2^{−1}$
Use $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$4 \cdot \frac{1}{2^{1}} 4 \cdot \frac{1}{2^{1}}$
Simplify.	$22$

ⓑ	${(4 \cdot 2)}^{−1} {(4 \cdot 2)}^{−1}$
Simplify inside the parentheses first.	${(8)}^{−1} {(8)}^{−1}$
Use $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{8^{1}} \frac{1}{8^{1}}$
Simplify.	$\frac{1}{8} \frac{1}{8}$

Try It 10.129

Simplify:

ⓐ $6 \cdot 3^{−1}$
ⓑ ${(6 \cdot 3)}^{−1}$

Try It 10.130

Simplify:

ⓐ $8 \cdot 2^{−2}$
ⓑ ${(8 \cdot 2)}^{−2}$

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.

Example 10.66

Simplify: $x^{−6} . x^{−6} .$

Show solution

Solution

	$x^{−6} x^{−6}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{x^{6}} \frac{1}{x^{6}}$

Try It 10.131

Simplify: $y^{−7} . y^{−7} .$

Try It 10.132

Simplify: $z^{−8} . z^{−8} .$

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.

Example 10.67

Simplify:

ⓐ $5 y^{−1}$
ⓑ ${(5 y)}^{−1}$
ⓒ ${(−5 y)}^{−1}$

Show solution

Solution

ⓐ
Notice the exponent applies to just the base $y y$ .	$5 y^{−1} 5 y^{−1}$
Take the reciprocal of $y y$ and change the sign of the exponent.	$5 \cdot \frac{1}{y^{1}} 5 \cdot \frac{1}{y^{1}}$
Simplify.	$\frac{5}{y} \frac{5}{y}$

ⓑ
Here the parentheses make the exponent apply to the base $5 y 5 y$ .	${(5 y)}^{−1} {(5 y)}^{−1}$
Take the reciprocal of $5 y 5 y$ and change the sign of the exponent.	$\frac{1}{{(5 y)}^{1}} \frac{1}{{(5 y)}^{1}}$
Simplify.	$\frac{1}{5 y} \frac{1}{5 y}$

ⓒ
	${(−5 y)}^{−1} {(−5 y)}^{−1}$
The base is $- 5 y - 5 y$ . Take the reciprocal of $- 5 y - 5 y$ and change the sign of the exponent.	$\frac{1}{{(−5 y)}^{1}} \frac{1}{{(−5 y)}^{1}}$
Simplify.	$\frac{1}{−5 y} \frac{1}{−5 y}$
Use $\frac{a}{- b} = - \frac{a}{b} . \frac{a}{- b} = - \frac{a}{b} .$	$- \frac{1}{5 y} - \frac{1}{5 y}$

Try It 10.133

Simplify:

ⓐ $8 p^{−1}$
ⓑ ${(8 p)}^{−1}$
ⓒ ${(−8 p)}^{−1}$

Try It 10.134

Simplify:

ⓐ $11 q^{−1}$
ⓑ ${(11 q)}^{−1}$
ⓒ ${(−11 q)}^{−1}$

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\frac{a^{m}}{a^{n}} = a^{m - n}, \frac{a^{m}}{a^{n}} = a^{m - n},$ where $a \neq 0 a \neq 0$ and m and n are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$

Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Summary of Exponent Properties

If $a, b a, b$ are real numbers and $m, n m, n$ are integers, then

\begin{matrix} Product Property & a^{m} \cdot a^{n} = a^{m + n} \\ Power Property & {(a^{m})}^{n} = a^{m \cdot n} \\ Product to a Power Property & {(a b)}^{m} = a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0 \\ Zero Exponent Property & a^{0} = 1, a \neq 0 \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0 \\ Definition of Negative Exponent & a^{- n} = \frac{1}{a^{n}} \end{matrix} \begin{matrix} Product Property & a^{m} \cdot a^{n} = a^{m + n} \\ Power Property & {(a^{m})}^{n} = a^{m \cdot n} \\ Product to a Power Property & {(a b)}^{m} = a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} = a^{m - n}, \\ Zero Exponent Property & a^{0} = 1, \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, \\ Definition of Negative Exponent & a^{- n} = \frac{1}{a^{n}} \end{matrix}

Example 10.68

Simplify:

ⓐ $x^{−4} \cdot x^{6}$
ⓑ $y^{−6} \cdot y^{4}$
ⓒ $z^{−5} \cdot z^{−3}$

Show solution

Solution

ⓐ
	$x^{−4} \cdot x^{6} x^{−4} \cdot x^{6}$
Use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} . a^{m} \cdot a^{n} = a^{m + n} .$	$x^{−4 + 6} x^{−4 + 6}$
Simplify.	$x^{2} x^{2}$

ⓑ
	$y^{−6} \cdot y^{4} y^{−6} \cdot y^{4}$
The bases are the same, so add the exponents.	$y^{−6 + 4} y^{−6 + 4}$
Simplify.	$y^{−2} y^{−2}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{y^{2}} \frac{1}{y^{2}}$

ⓒ
	$z^{−5} \cdot z^{−3} z^{−5} \cdot z^{−3}$
The bases are the same, so add the exponents.	$z^{−5−3} z^{−5−3}$
Simplify.	$z^{−8} z^{−8}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$\frac{1}{z^{8}} \frac{1}{z^{8}}$

Try It 10.135

Simplify:

ⓐ $x^{−3} \cdot x^{7}$
ⓑ $y^{−7} \cdot y^{2}$
ⓒ $z^{−4} \cdot z^{−5}$

Try It 10.136

Simplify:

ⓐ $a^{−1} \cdot a^{6}$
ⓑ $b^{−8} \cdot b^{4}$
ⓒ $c^{−8} \cdot c^{−7}$

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

Example 10.69

Simplify: $(m^{4} n^{−3}) (m^{−5} n^{−2}) . (m^{4} n^{−3}) (m^{−5} n^{−2}) .$

Show solution

Solution

	$(m^{4} n^{−3}) (m^{−5} n^{−2}) (m^{4} n^{−3}) (m^{−5} n^{−2})$
Use the Commutative Property to get like bases together.	$m^{4} m^{−5} \cdot n^{−2} n^{−3} m^{4} m^{−5} \cdot n^{−2} n^{−3}$
Add the exponents for each base.	$m^{−1} \cdot n^{−5} m^{−1} \cdot n^{−5}$
Take reciprocals and change the signs of the exponents.	$\frac{1}{m^{1}} \cdot \frac{1}{n^{5}} \frac{1}{m^{1}} \cdot \frac{1}{n^{5}}$
Simplify.	$\frac{1}{m n^{5}} \frac{1}{m n^{5}}$

Try It 10.137

Simplify: $(p^{6} q^{−2}) (p^{−9} q^{−1}) . (p^{6} q^{−2}) (p^{−9} q^{−1}) .$

Try It 10.138

Simplify: $(r^{5} s^{−3}) (r^{−7} s^{−5}) . (r^{5} s^{−3}) (r^{−7} s^{−5}) .$

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.

Example 10.70

Simplify: $(2 x^{−6} y^{8}) (−5 x^{5} y^{−3}) . (2 x^{−6} y^{8}) (−5 x^{5} y^{−3}) .$

Show solution

Solution

	$(2 x^{−6} y^{8}) (−5 x^{5} y^{−3}) (2 x^{−6} y^{8}) (−5 x^{5} y^{−3})$
Rewrite with the like bases together.	$2 (−5) \cdot (x^{−6} x^{5}) \cdot (y^{8} y^{−3}) 2 (−5) \cdot (x^{−6} x^{5}) \cdot (y^{8} y^{−3})$
Simplify.	$−10 \cdot x^{−1} \cdot y^{5} −10 \cdot x^{−1} \cdot y^{5}$
Use the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$−10 \cdot \frac{1}{x^{1}} \cdot y^{5} −10 \cdot \frac{1}{x^{1}} \cdot y^{5}$
Simplify.	$\frac{−10 y^{5}}{x} \frac{−10 y^{5}}{x}$

Try It 10.139

Simplify: $(3 u^{−5} v^{7}) (−4 u^{4} v^{−2}) . (3 u^{−5} v^{7}) (−4 u^{4} v^{−2}) .$

Try It 10.140

Simplify: $(−6 c^{−6} d^{4}) (−5 c^{−2} d^{−1}) . (−6 c^{−6} d^{4}) (−5 c^{−2} d^{−1}) .$

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Example 10.71

Simplify: ${(k^{3})}^{−2} . {(k^{3})}^{−2} .$

Show solution

Solution

	${(k^{3})}^{−2} {(k^{3})}^{−2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} . {(a b)}^{m} = a^{m} b^{m} .$	$k^{3 (−2)} k^{3 (−2)}$
Simplify.	$k^{−6} k^{−6}$
Rewrite with a positive exponent.	$\frac{1}{k^{6}} \frac{1}{k^{6}}$

Try It 10.141

Simplify: ${(x^{4})}^{−1} . {(x^{4})}^{−1} .$

Try It 10.142

Simplify: ${(y^{2})}^{−2} . {(y^{2})}^{−2} .$

Example 10.72

Simplify: ${(5 x^{−3})}^{2} . {(5 x^{−3})}^{2} .$

Show solution

Solution

	${(5 x^{−3})}^{2} {(5 x^{−3})}^{2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} . {(a b)}^{m} = a^{m} b^{m} .$	$5^{2} {(x^{−3})}^{2} 5^{2} {(x^{−3})}^{2}$
Simplify $5^{2} 5^{2}$ and multiply the exponents of $x x$ using the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} . {(a^{m})}^{n} = a^{m \cdot n} .$	$25 k^{−6} 25 k^{−6}$
Rewrite $x^{−6} x^{−6}$ by using the definition of a negative exponent, $a^{- n} = \frac{1}{a^{n}} . a^{- n} = \frac{1}{a^{n}} .$	$25 \cdot \frac{1}{x^{6}} 25 \cdot \frac{1}{x^{6}}$
Simplify	$\frac{25}{x^{6}} \frac{25}{x^{6}}$

Try It 10.143

Simplify: ${(8 a^{−4})}^{2} . {(8 a^{−4})}^{2} .$

Try It 10.144

Simplify: ${(2 c^{−4})}^{3} . {(2 c^{−4})}^{3} .$

To simplify a fraction, we use the Quotient Property.

Example 10.73

Simplify: $\frac{r^{5}}{r^{−4}} . \frac{r^{5}}{r^{−4}} .$

Show solution

Solution


Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n} \frac{a^{m}}{a^{n}} = a^{m - n}$ .

Simplify.

Try It 10.145

Simplify: $\frac{x^{8}}{x^{−3}} . \frac{x^{8}}{x^{−3}} .$

Try It 10.146

Simplify: $\frac{y^{7}}{y^{−6}} . \frac{y^{7}}{y^{−6}} .$

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of $10 . 10 .$ We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.

Consider the numbers $40004000$ and $0.004 . 0.004 .$ We know that $40004000$ means $4 \times 1000 4$ and $0.004 0.004$ means $4 \times \frac{1}{1000} . 4$ If we write the $10001000$ as a power of ten in exponential form, we can rewrite these numbers in this way:

\begin{matrix} 4000 & 0.004 \\ 4 \times 1000 & 4 \times \frac{1}{1000} \\ 4 \times 10^{3} & 4 \times \frac{1}{10^{3}} \\ 4 \times 10^{−3} \end{matrix} \begin{matrix} 4000 & 0.004 \\ 4 & 4 \\ 4 & 4 \\ 4 \end{matrix}

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than $10, 10,$ and the second factor is a power of $1010$ written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

a \times 10^{n} a

where $a \geq 1 a \geq 1$ and $a < 10 a < 10$ and $n n$ is an integer.

It is customary in scientific notation to use $\times$ as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, “Moved the decimal point 3 places to the left.” On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, “Moved the decimal point 3 places to the right.”

In both cases, the decimal was moved $33$ places to get the first factor, $4, 4,$ by itself.

The power of $1010$ is positive when the number is larger than $1 : 4000 = 4 \times 10^{3} . 1 :$
The power of $1010$ is negative when the number is between $00$ and $1 : 0.004 = 4 \times 10^{- 3} . 1 :$

Example 10.74

Write $37,000 37,000$ in scientific notation.

Show solution

Solution

Step 1: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Step 2: Count the number of decimal places, $n n$ , that the decimal point was moved.	3.70000 4 places
Step 3: Write the number as a product with a power of 10.	$3.7 \times 10^{4} 3.7 \times 10^{4}$
If the original number is: greater than 1, the power of 10 will be $10^{n} 10^{n}$ . between 0 and 1, the power of 10 will be $10^{−n} 10^{−n}$
Step 4: Check.
$10^{4} 10^{4}$ is 10,000 and 10,000 times 3.7 will be 37,000.
	$37,000 = 3.7 \times 10^{4} 37,000 = 3.7 \times 10^{4}$

Try It 10.147

Write in scientific notation: $96,000 . 96,000 .$

Try It 10.148

Write in scientific notation: $48,300 . 48,300 .$

How To

Convert from decimal notation to scientific notation.

Step 1. Move the decimal point so that the first factor is greater than or equal to $11$ but less than $10 . 10 .$
Step 2. Count the number of decimal places, $n, n,$ that the decimal point was moved.
Step 3.
Write the number as a product with a power of 10.10.
- If the original number is:
  - greater than $1, 1,$ the power of $1010$ will be $10^{n} . 10^{n} .$
  - between $00$ and $1, 1,$ the power of $1010$ will be $10^{- n} . 10^{- n} .$
Step 4. Check.

Example 10.75

Write in scientific notation: $0.0052 . 0.0052 .$

Show solution

Solution

	0.0052
Move the decimal point to get 5.2, a number between 1 and 10.
Count the number of decimal places the point was moved.	3 places
Write as a product with a power of 10.	$5.2 \times 10^{−3} 5.2 \times 10^{−3}$
Check your answer: $\begin{matrix} 5.2 \times 10^{−3} \\ 5.2 \times \frac{1}{10^{3}} \\ 5.2 \times \frac{1}{1000} \\ 5.2 \times 0.001 \\ 0.0052 \end{matrix} \begin{matrix} 5.2 \\ 5.2 \\ 0.0052 \end{matrix}$
	$0.0052 = 5.2 \times 10^{−3} 0.0052 = 5.2$

Try It 10.149

Write in scientific notation: $0.0078 . 0.0078 .$

Try It 10.150

Write in scientific notation: $0.0129 . 0.0129 .$

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

\begin{matrix} 9.12 \times 10^{4} & 9.12 \times 10^{−4} \\ 9.12 \times 10,000 & 9.12 \times 0.0001 \\ 91,200 & 0.000912 \end{matrix} \begin{matrix} 9.12 & 9.12 \\ 9.12 & 9.12 \\ 91,200 & 0.000912 \end{matrix}

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Example 10.76

Convert to decimal form: $6.2 \times 10^{3} . 6.2$

Show solution

Solution

Step 1: Determine the exponent, $n n$ , on the factor 10.	$6.2 \times 10^{3} 6.2 \times 10^{3}$
Step 2: Move the decimal point $n n$ places, adding zeros if needed.
If the exponent is positive, move the decimal point $n n$ places to the right. If the exponent is negative, move the decimal point $\| n \| \| n \|$ places to the left.	6,200
Step 3: Check to see if your answer makes sense.
$10^{3} 10^{3}$ is 1000 and 1000 times 6.2 will be 6,200.	$6.2 \times 10^{3} = 6,200 6.2 \times 10^{3} = 6,200$

Try It 10.151

Convert to decimal form: $1.3 \times 10^{3} . 1.3$

Try It 10.152

Convert to decimal form: $9.25 \times 10^{4} . 9.25$

How To

Convert scientific notation to decimal form.

Step 1. Determine the exponent, $n, n,$ on the factor $10 . 10 .$
Step 2.
Move the decimal nn places, adding zeros if needed.
- If the exponent is positive, move the decimal point $n n$ places to the right.
- If the exponent is negative, move the decimal point $| n | | n |$ places to the left.
Step 3. Check.

Example 10.77

Convert to decimal form: $8.9 \times 10^{−2} . 8.9$

Show solution

Solution

	$8.9 \times 10^{−2} 8.9 \times 10^{−2}$
Determine the exponent $n n$ , on the factor 10.	The exponent is −2.
Move the decimal point 2 places to the left.
Add zeros as needed for placeholders.	0.089
	$8.9 \times 10^{−2} = 0.089 8.9 \times 10^{−2} = 0.089$
The Check is left to you.

Try It 10.153

Convert to decimal form: $1.2 \times 10^{−4} . 1.2$

Try It 10.154

Convert to decimal form: $7.5 \times 10^{−2} . 7.5$

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Example 10.78

Multiply. Write answers in decimal form: $(4 \times 10^{5}) (2 \times 10^{−7}) . (4) (2) .$

Show solution

Solution

	$(4 \times 10^{5}) (2 \times 10^{−7}) (4) (2)$
Use the Commutative Property to rearrange the factors.	$4 \cdot 2 \cdot 10^{5} \cdot 10^{−7} 4 \cdot 2 \cdot 10^{5} \cdot 10^{−7}$
Multiply 4 by 2 and use the Product Property to multiply $10^{5} 10^{5}$ by $10^{−7} 10^{−7}$ .	$8 \times 10^{−2} 8$
Change to decimal form by moving the decimal two places left.	$0.08 0.08$

Try It 10.155

Multiply. Write answers in decimal form: $(3 \times 10^{6}) (2 \times 10^{−8}) . (3) (2) .$

Try It 10.156

Multiply. Write answers in decimal form: $(3 \times 10^{−2}) (3 \times 10^{−1}) . (3) (3) .$

Example 10.79

Divide. Write answers in decimal form: $\frac{9 \times 10^{3}}{3 \times 10^{−2}} . \frac{9}{3} .$

Show solution

Solution

	$\frac{9 \times 10^{3}}{3 \times 10^{−2}} \frac{9}{3}$
Separate the factors.	$\frac{9}{3} \times \frac{10^{3}}{10^{−2}} \frac{9}{3}$
Divide 9 by 3 and use the Quotient Property to divide $10^{3} 10^{3}$ by $10^{−2} 10^{−2}$ .	$3 \times 10^{5} 3$
Change to decimal form by moving the decimal five places right.	$300,000 300,000$

Try It 10.157

Divide. Write answers in decimal form: $\frac{8 \times 10^{4}}{2 \times 10^{−1}} . \frac{8}{2} .$

Try It 10.158

Divide. Write answers in decimal form: $\frac{8 \times 10^{2}}{4 \times 10^{−2}} . \frac{8}{4} .$

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Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

316.

$5^{−3} 5^{−3}$

317.

$8^{−2} 8^{−2}$

318.

$3^{−4} 3^{−4}$

319.

$2^{−5} 2^{−5}$

320.

$7^{−1} 7^{−1}$

321.

$10^{−1} 10^{−1}$

322.

$2^{−3} + 2^{−2} 2^{−3} + 2^{−2}$

323.

$3^{−2} + 3^{−1} 3^{−2} + 3^{−1}$

324.

$3^{−1} + 4^{−1} 3^{−1} + 4^{−1}$

325.

$10^{−1} + 2^{−1} 10^{−1} + 2^{−1}$

326.

$10^{0} - 10^{−1} + 10^{−2} 10^{0} - 10^{−1} + 10^{−2}$

327.

$2^{0} - 2^{−1} + 2^{−2} 2^{0} - 2^{−1} + 2^{−2}$

328.

ⓐ ${(−6)}^{−2}$
ⓑ $- 6^{−2}$

329.

ⓐ ${(−8)}^{−2}$
ⓑ $- 8^{−2}$

330.

ⓐ ${(−10)}^{−4}$
ⓑ $- 10^{−4}$

331.

ⓐ ${(−4)}^{−6}$
ⓑ $- 4^{−6}$

332.

ⓐ $5 \cdot 2^{−1}$
ⓑ ${(5 \cdot 2)}^{−1}$

333.

ⓐ $10 \cdot 3^{−1}$
ⓑ ${(10 \cdot 3)}^{−1}$

334.

ⓐ $4 \cdot 10^{−3}$
ⓑ ${(4 \cdot 10)}^{−3}$

335.

ⓐ $3 \cdot 5^{−2}$
ⓑ ${(3 \cdot 5)}^{−2}$

336.

$n^{−4} n^{−4}$

337.

$p^{−3} p^{−3}$

338.

$c^{−10} c^{−10}$

339.

$m^{−5} m^{−5}$

340.

ⓐ $4 x^{−1}$
ⓑ ${(4 x)}^{−1}$
ⓒ ${(−4 x)}^{−1}$

341.

ⓐ $3 q^{−1}$
ⓑ ${(3 q)}^{−1}$
ⓒ ${(−3 q)}^{−1}$

342.

ⓐ $6 m^{−1}$
ⓑ ${(6 m)}^{−1}$
ⓒ ${(−6 m)}^{−1}$

343.

ⓐ $10 k^{−1}$
ⓑ ${(10 k)}^{−1}$
ⓒ ${(−10 k)}^{−1}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

344.

$p^{−4} \cdot p^{8} p^{−4} \cdot p^{8}$

345.

$r^{−2} \cdot r^{5} r^{−2} \cdot r^{5}$

346.

$n^{−10} \cdot n^{2} n^{−10} \cdot n^{2}$

347.

$q^{−8} \cdot q^{3} q^{−8} \cdot q^{3}$

348.

$k^{−3} \cdot k^{−2} k^{−3} \cdot k^{−2}$

349.

$z^{−6} \cdot z^{−2} z^{−6} \cdot z^{−2}$

350.

$a \cdot a^{−4} a \cdot a^{−4}$

351.

$m \cdot m^{−2} m \cdot m^{−2}$

352.

$p^{5} \cdot p^{−2} \cdot p^{−4} p^{5} \cdot p^{−2} \cdot p^{−4}$

353.

$x^{4} \cdot x^{−2} \cdot x^{−3} x^{4} \cdot x^{−2} \cdot x^{−3}$

354.

$a^{3} b^{−3} a^{3} b^{−3}$

355.

$u^{2} v^{−2} u^{2} v^{−2}$

356.

$(x^{5} y^{−1}) (x^{−10} y^{−3}) (x^{5} y^{−1}) (x^{−10} y^{−3})$

357.

$(a^{3} b^{−3}) (a^{−5} b^{−1}) (a^{3} b^{−3}) (a^{−5} b^{−1})$

358.

$(u v^{−2}) (u^{−5} v^{−4}) (u v^{−2}) (u^{−5} v^{−4})$

359.

$(p q^{−4}) (p^{−6} q^{−3}) (p q^{−4}) (p^{−6} q^{−3})$

360.

$(−2 r^{−3} s^{9}) (6 r^{4} s^{−5}) (−2 r^{−3} s^{9}) (6 r^{4} s^{−5})$

361.

$(−3 p^{−5} q^{8}) (7 p^{2} q^{−3}) (−3 p^{−5} q^{8}) (7 p^{2} q^{−3})$

362.

$(−6 m^{−8} n^{−5}) (−9 m^{4} n^{2}) (−6 m^{−8} n^{−5}) (−9 m^{4} n^{2})$

363.

$(−8 a^{−5} b^{−4}) (−4 a^{2} b^{3}) (−8 a^{−5} b^{−4}) (−4 a^{2} b^{3})$

364.

${(a^{3})}^{−3} {(a^{3})}^{−3}$

365.

${(q^{10})}^{−10} {(q^{10})}^{−10}$

366.

${(n^{2})}^{−1} {(n^{2})}^{−1}$

367.

${(x^{4})}^{−1} {(x^{4})}^{−1}$

368.

${(y^{−5})}^{4} {(y^{−5})}^{4}$

369.

${(p^{−3})}^{2} {(p^{−3})}^{2}$

370.

${(q^{−5})}^{−2} {(q^{−5})}^{−2}$

371.

${(m^{−2})}^{−3} {(m^{−2})}^{−3}$

372.

${(4 y^{−3})}^{2} {(4 y^{−3})}^{2}$

373.

${(3 q^{−5})}^{2} {(3 q^{−5})}^{2}$

374.

${(10 p^{−2})}^{−5} {(10 p^{−2})}^{−5}$

375.

${(2 n^{−3})}^{−6} {(2 n^{−3})}^{−6}$

376.

$\frac{u^{9}}{u^{−2}} \frac{u^{9}}{u^{−2}}$

377.

$\frac{b^{5}}{b^{−3}} \frac{b^{5}}{b^{−3}}$

378.

$\frac{x^{−6}}{x^{4}} \frac{x^{−6}}{x^{4}}$

379.

$\frac{m^{5}}{m^{−2}} \frac{m^{5}}{m^{−2}}$

380.

$\frac{q^{3}}{q^{12}} \frac{q^{3}}{q^{12}}$

381.

$\frac{r^{6}}{r^{9}} \frac{r^{6}}{r^{9}}$

382.

$\frac{n^{−4}}{n^{−10}} \frac{n^{−4}}{n^{−10}}$

383.

$\frac{p^{−3}}{p^{−6}} \frac{p^{−3}}{p^{−6}}$

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

384.

45,000

385.

280,000

386.

8,750,000

387.

1,290,000

388.

0.036

389.

0.041

390.

0.00000924

391.

0.0000103

392.

The population of the United States on July 4, 2010 was almost $310,000,000 . 310,000,000 .$

393.

The population of the world on July 4, 2010 was more than $6,850,000,000 . 6,850,000,000 .$

394.

The average width of a human hair is $0.0018 0.0018$ centimeters.

395.

The probability of winning the $20102010$ Megamillions lottery is about $0.0000000057 . 0.0000000057 .$

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

396.

$4.1 \times 10^{2} 4.1$

397.

$8.3 \times 10^{2} 8.3$

398.

$5.5 \times 10^{8} 5.5$

399.

$1.6 \times 10^{10} 1.6$

400.

$3.5 \times 10^{−2} 3.5$

401.

$2.8 \times 10^{−2} 2.8$

402.

$1.93 \times 10^{−5} 1.93$

403.

$6.15 \times 10^{−8} 6.15$

404.

In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was $2 \times 10^{4} . 2$

405.

At the start of 2012, the US federal budget had a deficit of more than $$1.5 \times 10^{13} . $1.5$

406.

The concentration of carbon dioxide in the atmosphere is $3.9 \times 10^{−4} . 3.9$

407.

The width of a proton is $1 \times 10^{−5} 1$ of the width of an atom.

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

408.

$(2 \times 10^{5}) (2 \times 10^{−9}) (2) (2)$

409.

$(3 \times 10^{2}) (1 \times 10^{−5}) (3) (1)$

410.

$(1.6 \times 10^{−2}) (5.2 \times 10^{−6}) (1.6) (5.2)$

411.

$(2.1 \times 10^{−4}) (3.5 \times 10^{−2}) (2.1) (3.5)$

412.

$\frac{6 \times 10^{4}}{3 \times 10^{−2}} \frac{6}{3}$

413.

$\frac{8 \times 10^{6}}{4 \times 10^{−1}} \frac{8}{4}$

414.

$\frac{7 \times 10^{−2}}{1 \times 10^{−8}} \frac{7}{1}$

415.

$\frac{5 \times 10^{−3}}{1 \times 10^{−10}} \frac{5}{1}$

Everyday Math

416.

Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by $1.5 1.5$ trillion calories by the end of 2015.

ⓐ Write $1.5 1.5$ trillion in decimal notation.
ⓑ Write $1.5 1.5$ trillion in scientific notation.

417.

Length of a year The difference between the calendar year and the astronomical year is $0.000125 0.000125$ day.

ⓐ Write this number in scientific notation.
ⓑ How many years does it take for the difference to become 1 day?

418.

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided $11$ by $2,598,960 2,598,960$ and saw the answer $3.848 \times 10^{−7} . 3.848$ Write the number in decimal notation.

419.

Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with $66$ of her $5050$ favorite photographs, she multiplied $50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 . 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46 \cdot 45 .$ Her calculator gave the answer $1.1441304 \times 10^{10} . 1.1441304$ Write the number in decimal notation.

Writing Exercises

420.

ⓐ Explain the meaning of the exponent in the expression $2^{3} . 2^{3} .$
ⓑ Explain the meaning of the exponent in the expression $2^{−3} 2^{−3}$

421.

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?