Integer Exponents and Scientific Notation

6.7 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 6.15

Before you get started, take this readiness quiz.

What is the place value of the $6$ in the number $64,891$ ?
If you missed this problem, review Example 1.1.

Be Prepared 6.16

Name the decimal: $0.0012 .$
If you missed this problem, review Example 1.91.

Be Prepared 6.17

Subtract: $5 - (−3) .$
If you missed this problem, review Example 1.42.

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

Quotient Property for Exponents

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

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

What if we just subtract exponents regardless of which is larger?

Let’s consider $\frac{x^{2}}{x^{5}}$ .

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

\begin{matrix} \frac{x^{2}}{x^{5}} \\ x^{2 - 5} \\ x^{−3} \end{matrix}

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

Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.

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

Negative Exponent

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

The negative exponent tells us we can 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 the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression $x^{−3}$ , we will take one more step and write $\frac{1}{x^{3}}$ . The answer is considered to be in simplest form when it has only positive exponents.

Example 6.89

Simplify: ⓐ $4^{−2}$ ⓑ $10^{−3} .$

Solution

ⓐ	$4^{−2}$
Use the definition of a negative exponent, $a^{− n} = \frac{1}{a^{n}}$ .	$\frac{1}{4^{2}}$
Simplify.	$\frac{1}{16}$
ⓑ	$10^{−3}$
Use the definition of a negative exponent, $a^{− n} = \frac{1}{a^{n}}$ .	$\frac{1}{10^{3}}$
Simplify.	$\frac{1}{1000}$

Step-by-step examples demonstrating the simplification of mathematical expressions with negative exponents.

Try It 6.177

Simplify: ⓐ $2^{−3}$ ⓑ $10^{−7} .$

Try It 6.178

Simplify: ⓐ $3^{−2}$ ⓑ $10^{−4} .$

In Example 6.89 we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

	$\frac{1}{a^{− n}}$
Use the definition of a negative exponent, $a^{− n} = \frac{1}{a^{n}}$ .	$\frac{1}{\frac{1}{a^{n}}}$
Simplify the complex fraction.	$1 \cdot \frac{a^{n}}{1}$
Multiply.	$a^{n}$

Steps demonstrating the simplification of the expression 1/(a^-n) using the definition of negative exponents, resulting in a^n.

This leads to the Property of Negative Exponents.

Property of Negative Exponents

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

Example 6.90

Simplify: ⓐ $\frac{1}{y^{−4}}$ ⓑ $\frac{1}{3^{−2}} .$

Solution

ⓐ	$\frac{1}{y^{−4}}$
Use the property of a negative exponent, $\frac{1}{a^{− n}} = a^{n}$ .	$y^{4}$
ⓑ	$\frac{1}{3^{−2}}$
Use the property of a negative exponent, $\frac{1}{a^{− n}} = a^{n}$ .	$3^{2}$
Simplify.	$9$

Examples demonstrating step-by-step simplification of mathematical expressions using the property of negative exponents.

Try It 6.179

Simplify: ⓐ $\frac{1}{p^{−8}}$ ⓑ $\frac{1}{4^{−3}} .$

Try It 6.180

Simplify: ⓐ $\frac{1}{q^{−7}}$ ⓑ $\frac{1}{2^{−4}} .$

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

	${(\frac{3}{4})}^{−2}$
Use the definition of a negative exponent, $a^{− n} = \frac{1}{a^{n}}$ .	$\frac{1}{{(\frac{3}{4})}^{2}}$
Simplify the denominator.	$\frac{1}{\frac{9}{16}}$
Simplify the complex fraction.	$\frac{16}{9}$
But we know that $\frac{16}{9}$ is ${(\frac{4}{3})}^{2}$ .
This tells us that:	${(\frac{3}{4})}^{−2} = {(\frac{4}{3})}^{2}$

Table 6.4 Illustrates the step-by-step simplification of a fractional expression with a negative exponent, demonstrating the rule for inverses.

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Exponent Property

If $a and b$ are real numbers, $a \neq 0, b \neq 0,$ and $n$ is an integer, then ${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$ .

Example 6.91

Simplify: ⓐ ${(\frac{5}{7})}^{−2}$ ⓑ ${(- \frac{2 x}{y})}^{−3} .$

Solution

ⓐ	${(\frac{5}{7})}^{−2}$
Use the Quotient to a Negative Exponent Property, ${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$ .
Take the reciprocal of the fraction and change the sign of the exponent.	${(\frac{7}{5})}^{2}$
Simplify.	$\frac{49}{25}$
ⓑ	${(- \frac{2 x}{y})}^{−3}$
Use the Quotient to a Negative Exponent Property, ${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$ .
Take the reciprocal of the fraction and change the sign of the exponent.	${(- \frac{y}{2 x})}^{3}$
Simplify.	$- \frac{y^{3}}{8 x^{3}}$

Step-by-step simplification of expressions using the Quotient to a Negative Exponent Property.

Try It 6.181

Simplify: ⓐ ${(\frac{2}{3})}^{−4}$ ⓑ ${(- \frac{6 m}{n})}^{−2} .$

Try It 6.182

Simplify: ⓐ ${(\frac{3}{5})}^{−3}$ ⓑ ${(- \frac{a}{2 b})}^{−4} .$

When simplifying an expression with exponents, we must be careful to correctly identify the base.

Example 6.92

Simplify: ⓐ ${(−3)}^{−2}$ ⓑ $− 3^{−2}$ ⓒ ${(- \frac{1}{3})}^{−2}$ ⓓ $− {(\frac{1}{3})}^{−2} .$

Solution

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

Step-by-step evaluation of mathematical expressions involving negative exponents, demonstrating their application to various bases and handling of negative signs.

Try It 6.183

Simplify: ⓐ ${(−5)}^{−2}$ ⓑ $− 5^{−2}$ ⓒ ${(- \frac{1}{5})}^{−2}$ ⓓ $− {(\frac{1}{5})}^{−2} .$

Try It 6.184

Simplify: ⓐ ${(−7)}^{−2}$ ⓑ $− 7^{−2}$ , ⓒ ${(- \frac{1}{7})}^{−2}$ ⓓ $− {(\frac{1}{7})}^{−2} .$

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

Example 6.93

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

Solution

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

Step-by-step simplification of expressions involving exponents and multiplication, demonstrating the order of operations.

Try It 6.185

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

Try It 6.186

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. We will assume all variables are non-zero.

Example 6.94

Simplify: ⓐ $x^{−6}$ ⓑ ${(u^{4})}^{−3} .$

Solution

ⓐ
$\begin{matrix} x^{−6} \\ Use the definition of a negative exponent, a^{− n} = \frac{1}{a^{n}} . & \frac{1}{x^{6}} \end{matrix}$
ⓑ
$\begin{matrix} {(u^{4})}^{−3} \\ Use the definition of a negative exponent, a^{− n} = \frac{1}{a^{n}} . & \frac{1}{{(u^{4})}^{3}} \\ Simplify. & \frac{1}{u^{12}} \end{matrix}$

Try It 6.187

Simplify: ⓐ $y^{−7}$ ⓑ ${(z^{3})}^{−5} .$

Try It 6.188

Simplify: ⓐ $p^{−9}$ ⓑ ${(q^{4})}^{−6} .$

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, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.

Example 6.95

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

Solution

ⓐ

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

The table demonstrates the steps to simplify the expression 5y⁻¹, illustrating the rule for negative exponents by taking the reciprocal.

ⓑ

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

Steps to simplify an algebraic expression with a negative exponent, detailing each transformation.

ⓒ

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

Step-by-step simplification of the algebraic expression (-5y)^-1.

Try It 6.189

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

Try It 6.190

Simplify: ⓐ ${11 q}^{−1}$ ⓑ ${(11 q)}^{−1}$ $− {(11 q)}^{−1}$ ⓒ $- \frac{1}{11 q}$

With negative exponents, the Quotient Rule needs only one form $\frac{a^{m}}{a^{n}} = a^{m - n}$ , for $a \neq 0$ . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.

Simplify Expressions with Integer Exponents

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

Summary of Exponent Properties

If $a and b$ are real numbers, and $m and n$ 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 & {(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 \\ Properties of Negative Exponents & a^{− n} & = & \frac{1}{a^{n}} and \frac{1}{a^{− n}} = a^{n} \\ Quotient to a Negative Exponent & {(\frac{a}{b})}^{− n} & = & {(\frac{b}{a})}^{n} \end{matrix}

Example 6.96

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

Solution

ⓐ

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

Steps to simplify the exponential expression x^-4 * x^6 using the product property of exponents.

ⓑ

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

Step-by-step simplification of an algebraic expression involving negative exponents using exponent rules.

ⓒ

	$z^{−5} \cdot z^{−3}$
Add the exponents, since the bases are the same.	$z^{−5 - 3}$
Simplify.	$z^{−8}$
Take the reciprocal and change the sign of the exponent, using the definition of a negative exponent.	$\frac{1}{z^{8}}$

Step-by-step simplification of an expression involving negative exponents: z^-5 * z^-3.

Try It 6.191

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

Try It 6.192

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.

Example 6.97

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

Solution

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

Step-by-step guide to simplifying an algebraic expression with negative exponents using exponent properties.

Try It 6.193

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

Try It 6.194

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

Try It 6.195

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

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

Example 6.98

Simplify: ${(6 k^{3})}^{−2} .$

Solution

	${(6 k^{3})}^{−2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	${(6)}^{−2} {(k^{3})}^{−2}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$6^{−2} k^{−6}$
Use the Definition of a Negative Exponent, $a^{− n} = \frac{1}{a^{n}} .$	$\frac{1}{6^{2}} \cdot \frac{1}{k^{6}}$
Simplify.	$\frac{1}{36 k^{6}}$

Demonstration of simplifying the algebraic expression (6k^3)^-2 using various exponent properties.

Try It 6.196

Simplify: ${(−4 x^{4})}^{−2} .$

Try It 6.197

Simplify: ${(2 b^{3})}^{−4} .$

Example 6.99

Simplify: ${(5 x^{−3})}^{2} .$

Solution

	${(5 x^{−3})}^{2}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$5^{2} {(x^{−3})}^{2}$
Simplify 5² and multiply the exponents of x using the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$25 \cdot x^{−6}$
Rewrite x⁻⁶ by using the Definition of a Negative Exponent, $a^{− n} = \frac{1}{a^{n}} .$	$25 \cdot \frac{1}{x^{6}}$
Simplify.	$\frac{25}{x^{6}}$

Step-by-step simplification of an algebraic expression using exponent properties.

Try It 6.198

Simplify: ${(8 a^{−4})}^{2} .$

Try It 6.199

Simplify: ${(2 c^{−4})}^{3} .$

To simplify a fraction, we use the Quotient Property and subtract the exponents.

Example 6.100

Simplify: $\frac{r^{5}}{r^{−4}} .$

Solution

	$\frac{r^{5}}{r^{−4}}$
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n} .$	$r^{5 - (−4)}$
Simplify.	$r^{9}$

Step-by-step simplification of an exponential expression, r^5 / r^-4, using the Quotient Property of Exponents.

Try It 6.200

Simplify: $\frac{x^{8}}{x^{−3}} .$

Try It 6.201

Simplify: $\frac{y^{8}}{y^{−6}} .$

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. 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 4,000 and $0.004$ . We know that 4,000 means $4 \times 1,000$ and 0.004 means $4 \times \frac{1}{1,000}$ .

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

\begin{matrix} 4,000 & 0.004 \\ 4 \times 1,000 & 4 \times \frac{1}{1,000} \\ 4 \times 10^{3} & 4 \times \frac{1}{10^{3}} \\ 4 \times 10^{−3} \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, and the second factor is a power of 10 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

\begin{matrix} a \times 10^{n} where 1 \leq │ a │ < 10 and n is an integer \end{matrix}

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

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

This figure illustrates how to convert a number to scientific notation. It has two columns. In the first column is 4000 equals 4 times 10 to the third power. Below this, the equation is repeated, with an arrow demonstrating that the decimal point at the end of 4000 has moved three places to the left, so that 4000 becomes 4.000. The second column has 0.004 equals 4 times 10 to the negative third power. Below this, the equation is repeated, with an arrow demonstrating how the decimal point in 0.004 is moved three places to the right to produce 4.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

$\begin{matrix} The power of 10 is positive when the number is larger than 1: & 4,000 = 4 \times 10^{3} \\ The power of 10 is negative when the number is between 0 and 1: & 0.004 = 4 \times 10^{−3} \end{matrix}$

Example 6.101

How to Convert from Decimal Notation to Scientific Notation

Write in scientific notation: 37,000.

Solution

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.” The second cell reads “Remember, there is a decimal at the end of 37,000.” The third cell contains 37,000. One line down, the second cell reads “Move the decimal after the 3. 3.7000 is between 1 and 10.”

In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7.

In the third row, the first cell reads “Step 3. Write the number as a product with a power of 10. If the original number is greater than 1, the power of 10 will be 10 to the n power. If it’s between 0 and 1, the power of 10 will be 10 to the negative n power.” The second cell reads “37,000 is greater than 1, so the power of 10 will have exponent 4.” The third cell contains 3.7 times 10 to the fourth power.

In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.

Try It 6.202

Write in scientific notation: $96,000 .$

Try It 6.203

Write in scientific notation: $48,300 .$

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 1 but less than 10.
Step 2. Count the number of decimal places, n, that the decimal point was moved.
Step 3.
Write the number as a product with a power of 10.
If the original number is:
- greater than 1, the power of 10 will be 10ⁿ.
- between 0 and 1, the power of 10 will be 10⁻ⁿ.
Step 4. Check.

Example 6.102

Write in scientific notation: $0.0052 .$

Solution

The original number, $0.0052$ , is between 0 and 1 so we will have a negative power of 10.


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

Try It 6.204

Write in scientific notation: $0.0078 .$

Try It 6.205

Write in scientific notation: $0.0129 .$

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}

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.

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

This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”

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 6.103

How to Convert Scientific Notation to Decimal Form

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

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.

In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.

In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.

Try It 6.206

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

Try It 6.207

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

The steps are summarized below.

How To

Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

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

Example 6.104

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

Solution


Determine the exponent, n, on the factor 10.
Since the exponent is negative, move the decimal point 2 places to the left.
Add zeros as needed for placeholders.

Try It 6.208

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

Try It 6.209

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

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Example 6.105

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

Solution

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

Try It 6.210

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

Try It 6.211

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

Example 6.106

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

Solution

	$\frac{9 \times 10^{3}}{3 \times 10^{−2}}$
Separate the factors, rewriting as the product of two fractions.	$\frac{9}{3} \times \frac{10^{3}}{10^{−2}}$
Divide.	$3 \times 10^{5}$
Change to decimal form by moving the decimal five places right.	300,000

Illustrates the step-by-step division of numbers in scientific notation, converting the final answer to decimal form.

Try It 6.212

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

Try It 6.213

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

Media

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

Section 6.7 Exercises

Practice Makes Perfect

Use the Definition of a Negative Exponent

In the following exercises, simplify.

500.

ⓐ $4^{−2}$ ⓑ $10^{−3}$

501.

ⓐ $3^{−4}$ ⓑ $10^{−2}$

502.

ⓐ $5^{- 3}$ ⓑ $10^{−5}$

503.

ⓐ $2^{−8}$ ⓑ $10^{−2}$

504.

ⓐ $\frac{1}{c^{−5}}$ ⓑ $\frac{1}{3^{−2}}$

505.

ⓐ $\frac{1}{c^{−5}}$ ⓑ $\frac{1}{5^{−2}}$

506.

ⓐ $\frac{1}{q^{−10}}$ ⓑ $\frac{1}{10^{−3}}$

507.

ⓐ $\frac{1}{t^{−9}}$ ⓑ $\frac{1}{10^{−4}}$

508.

ⓐ ${(\frac{5}{8})}^{−2}$ ⓑ ${(- \frac{3 m}{n})}^{−2}$

509.

ⓐ ${(\frac{3}{10})}^{−2}$ ⓑ ${(- \frac{2}{c d})}^{−3}$

510.

ⓐ ${(\frac{4}{9})}^{−3}$ ⓑ ${(- \frac{u^{2}}{2 v})}^{−5}$

511.

ⓐ ${(\frac{7}{2})}^{−3}$ ⓑ ${(- \frac{3}{x y^{2}})}^{−3}$

512.

ⓐ ${(- 5)}^{- 2}$ ⓑ $− 5^{−2}$ ⓒ ${(- \frac{1}{5})}^{−2}$ ⓓ $− {(\frac{1}{5})}^{−2}$

513.

ⓐ ${(−7)}^{−2}$ ⓑ $- 7^{- 2}$ ⓒ ${(- \frac{1}{7})}^{−2}$ ⓓ $− {(\frac{1}{7})}^{−2}$

514.

ⓐ $− 3^{−3}$ ⓑ ${(- \frac{1}{3})}^{−3}$ ⓒ $− {(\frac{1}{3})}^{−3}$ ⓓ ${(−3)}^{−3}$

515.

ⓐ $− 5^{−3}$ ⓑ ${(- \frac{1}{5})}^{−3}$ ⓒ $− {(\frac{1}{5})}^{−3}$ ⓓ ${(−5)}^{−3}$

516.

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

517.

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

518.

ⓐ $4 \cdot 5^{−2}$ ⓑ ${(4 \cdot 5)}^{−2}$

519.

ⓐ $3 \cdot 4^{−2}$ ⓑ ${(3 \cdot 4)}^{−2}$

520.

ⓐ $m^{−4}$ ⓑ ${(x^{3})}^{−4}$

521.

ⓐ $b^{−5}$ ⓑ ${(k^{2})}^{−5}$

522.

ⓐ $p^{−10}$ ⓑ ${(q^{6})}^{−8}$

523.

ⓐ $s^{−8}$ ⓑ ${(a^{9})}^{−10}$

524.

ⓐ $7 n^{−1}$ ⓑ ${(7 n)}^{−1}$ ⓒ ${(−7 n)}^{−1}$

525.

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

526.

ⓐ ${(3 p)}^{−2}$ ⓑ $3 p^{−2}$ ⓒ $−3 p^{−2}$

527.

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

528.

529.

530.

531.

532.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

533.

$x^{4} \cdot x^{−2} \cdot x^{−3}$

534.

$(w^{4} x^{−5}) (w^{−2} x^{−4})$

535.

$(m^{3} n^{−3}) (m^{−5} n^{−1})$

536.

$(u v^{−2}) (u^{−5} v^{−3})$

537.

$(p q^{−4}) (p^{−6} q^{−3})$

538.

$(−6 c^{−3} d^{9}) (2 c^{4} d^{−5})$

539.

$(−2 j^{−5} k^{8}) (7 j^{2} k^{−3})$

540.

$(−4 r^{−2} s^{−8}) (9 r^{4} s^{3})$

541.

$(−5 m^{4} n^{6}) (8 m^{−5} n^{−3})$

542.

${(5 x^{2})}^{−2}$

543.

${(4 y^{3})}^{−3}$

544.

${(3 z^{−3})}^{2}$

545.

${(2 p^{−5})}^{2}$

546.

$\frac{t^{9}}{t^{−3}}$

547.

$\frac{n^{5}}{n^{−2}}$

548.

$\frac{x^{−7}}{x^{−3}}$

549.

$\frac{y^{−5}}{y^{−10}}$

Convert from Decimal Notation to Scientific Notation

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

550.

57,000

551.

340,000

552.

8,750,000

553.

1,290,000

554.

0.026

555.

0.041

556.

0.00000871

557.

0.00000103

Convert Scientific Notation to Decimal Form

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

558.

$5.2 \times 10^{2}$

559.

$8.3 \times 10^{2}$

560.

$7.5 \times 10^{6}$

561.

$1.6 \times 10^{10}$

562.

$2.5 \times 10^{−2}$

563.

$3.8 \times 10^{−2}$

564.

$4.13 \times 10^{−5}$

565.

$1.93 \times 10^{−5}$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

566.

$(3 \times 10^{−5}) (3 \times 10^{9})$

567.

$(2 \times 10^{2}) (1 \times 10^{−4})$

568.

$(7.1 \times 10^{−2}) (2.4 \times 10^{−4})$

569.

$(3.5 \times 10^{−4}) (1.6 \times 10^{−2})$

In the following exercises, divide. Write your answer in decimal form.

570.

$\frac{7 \times 10^{−3}}{1 \times 10^{−7}}$

571.

$\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

572.

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

573.

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

Everyday Math

574.

The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation.

575.

The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation

576.

The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation.

577.

The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.

578.

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

579.

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

580.

The concentration of carbon dioxide in the atmosphere is $3.9 \times 10^{−4}$ . Convert this number to decimal form.

581.

The width of a proton is $1 \times 10^{−5}$ of the width of an atom. Convert this number to decimal form.

582.

Health care costs The Centers for Medicare and Medicaid projects that consumers will spend more than $4 trillion on health care by 2017.

ⓐ Write 4 trillion in decimal notation.
ⓑ Write 4 trillion in scientific notation.

583.

Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

584.

Distance The distance between Earth and one of the brightest stars in the night star is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion), miles.

ⓐ Write the number of miles in one light year in scientific notation.
ⓑ Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation.

585.

Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.

ⓐ Write the debt in scientific notation.
ⓑ Write the population in scientific notation.
ⓒ Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.

Writing Exercises

586.

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

587.

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.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “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,” and “multiply and divide using scientific notation.” The rest of the cells are blank.

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