5.2 Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression $a^{m},$ the exponent m tells us how many times we use the base a as a factor.

First example: a raised to the power of m equals a times a times a times a and so on until you have multiplied m different factors of a together. Second example: the quantity negative 9 raised to the power of 5 equals negative 9 times negative 9 times negative 9 times negative 9 times negative 9, a total of 5 factors of negative 9.

Let’s review the vocabulary for expressions with exponents.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.


What does this mean?

Notice that 5 is the sum of the exponents, 2 and 3. We see $x^{2} \cdot x^{3}$ is $x^{2 + 3}$ or $x to the power 5.$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Example 1

Simplify each expression: ⓐ $y^{5} \cdot y^{6}$ ⓑ $2^{x} \cdot 2^{3 x}$ ⓒ $2 a^{7} \cdot 3 a .$ ⓓ $d^{4} \cdot d^{5} \cdot d^{2}$

ⓐ


Use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.

ⓑ


Use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.

ⓒ


Rewrite, $a = a^{1} .$
Use the Commutative Property and use the Product Property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.

ⓓ


Add the exponents, since bases are the same.
Simplify.

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

Consider	$\frac{x^{5}}{x^{2}}$	$\frac{x^{2}}{x^{3}}$
What do they mean?	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$	$\frac{x \cdot x}{x \cdot x \cdot x}$
Use the Equivalent Fractions Property.	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$	$\frac{x \cdot x \cdot 1}{x \cdot x \cdot x}$
Simplify.	$x^{3}$	$\frac{1}{x}$

Notice, in each case the bases were the same and we subtracted exponents. We see $\frac{x^{5}}{x^{2}}$ is $x^{5 - 2}$ or $x^{3}$ . We see $\frac{x^{2}}{x^{3}}$ is or $\frac{1}{x} .$ When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator--notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. $\frac{x}{x} = 1$ . This leads to the Quotient Property for Exponents.

Example 2

Simplify each expression: ⓐ $\frac{x^{9}}{x^{7}}$ ⓑ $\frac{3^{10}}{3^{2}}$ ⓒ $\frac{b^{8}}{b^{12}}$ ⓓ $\frac{7^{3}}{7^{5}} .$

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

ⓐ

Since $9 greater than 7,$ there are more factors of $x$ in the numerator.
Use Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n} .$
Simplify.

ⓑ

Since $10 greater than 2,$ there are more factors of $3$ in the numerator.	$A mathematical fraction showing 3 to the power of 10 divided by 3 to the power of 2, an example of exponent rules where bases are the same.$
Use Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n} .$
Simplify.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

ⓒ

Since $12 greater than 8,$ there are more factors of $b$ in the denominator.
Use Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} .$	$A mathematical expression showing the fraction 1 over b to the power of (12 minus 8).$
Simplify.	$A mathematical expression showing the fraction one over b to the power of four, or 1/b^4.$

ⓓ

Since $5 greater than 3,$ there are more factors of $3$ in the denominator.
Use Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} .$	$A mathematical expression displaying the fraction 1 over 7 raised to the power of (s minus 3). The numerator '1' is in red, and the exponent 's-3' is also in a reddish-brown color.$
Simplify.	$A fraction with 1 in the numerator and 7 squared (7^2) in the denominator, representing 1/49.$
Simplify.	$The fraction one over forty-nine, represented numerically as 1/49, is displayed against a white background.$

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\frac{a^{m}}{a^{m}} .$ We know $, \frac{x}{x} = 1,$ for any $x open parenthesis x not equal to 0 close parenthesis$ since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify $\frac{a^{m}}{a^{m}} .$ when $m > n$ and when $n < m$ by subtracting exponents. What if $m = n ?$ We will simplify $\frac{a^{m}}{a^{m}}$ in two ways to lead us to the definition of the Zero Exponent Property. In general, for $a not equal to 0:$

In the first way we write a to the power of m divided by a to the power of m as a to the power of the quantity m minus m. This is equal to a to the power of 0. In the second way we write a to the power of m divided by a to the power of m as a fraction with m factors of a in the numerator and a factors of m in the denominator. Simplifying this we can cross of all the factors and are left with the number 1. This shows that a to the power of 0 is equal to 1.

We see $\frac{a^{m}}{a^{m}}$ simplifies to $a to the power 0$ and to 1. So $a^{0} = 1 .$ Any non-zero base raised to the power of zero equals 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. 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. We see $\frac{x^{2}}{x^{5}}$ is $x^{2 - 5}$ or $x to the power −3.$

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

In the figure the expression x raised to the power of 2 divided by x raised to the power of 5 is written as a fraction with 2 factors of x in the numerator divided by 5 factors of x in the denominator. Two factors are crossed off in both the numerator and denominator. This only leaves 3 factors of x in the denominator. The simplified fraction is 1 divided by x to the power of 3.

This implies that $x^{−3} = \frac{1}{x^{3}}$ and it leads us to the definition of a negative exponent. If n is an integer and $a not equal to 0,$ then $a^{− n} = \frac{1}{a^{n}} .$

Let’s now look 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.	$a^{n}$

This implies $\frac{1}{a^{− n}} = a^{n}$ and is another form of the definition of Properties of Negative Exponents.

The negative exponent tells us we can rewrite 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 $x to the power −3,$ 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 4

Simplify each expression: ⓐ $x to the power −5$ ⓑ $10 to the power −3$ ⓒ $\frac{1}{y^{−4}}$ ⓓ $\frac{1}{3^{−2}} .$

ⓐ

	$x to the power −5$
Use the definition of a negative exponent, $a^{− n} = \frac{1}{a^{n}} .$	$\frac{1}{x^{5}}$

ⓑ

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

ⓒ

	$\frac{1}{y^{−4}}$
Use the property of a negative exponent, $\frac{1}{a^{− n}} = a^{n} .$	$y to the power 4$

ⓓ

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

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	${(\frac{3}{4})}^{−2} = {(\frac{4}{3})}^{2}$

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.

Example 5

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

ⓐ

	${(\frac{5}{7})}^{−2}$
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.	$\frac{49}{25}$

ⓑ

	${(- \frac{x}{y})}^{−3}$
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.	${(- \frac{y}{x})}^{3}$
Simplify.	$- \frac{y^{3}}{x^{3}}$

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Example 6

Simplify each expression: ⓐ $z^{−5} \cdot z^{−3}$ ⓑ $(m^{4} n^{−3}) (m^{−5} n^{−2})$ ⓒ $(2 x^{−6} y^{8}) (−5 x^{5} y^{−3}) .$

ⓐ

	$z^{−5} \cdot z^{−3}$
Add the exponents, since the bases are the same.	$z^{−5 - 3}$
Simplify.	$z to the power −8$
Use the definition of a negative exponent.	$\frac{1}{z^{8}}$

ⓑ

	$(m^{4} n^{−3}) (m^{−5} n^{−2})$
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.	$\frac{1}{m n^{5}}$

ⓒ

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

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

$\begin{matrix} {(x^{2})}^{3} \\ What does this mean? & x^{2} \cdot x^{2} \cdot x^{2} \end{matrix}$

How many factors altogether?
So we have

Notice the 6 is the product of the exponents, 2 and 3. We see that ${(x^{2})}^{3}$ is $x^{2 \cdot 3}$ or $x to the power 6.$

We multiplied the exponents. This leads to the Power Property for Exponents.

Example 7

Simplify each expression: ⓐ ${(y^{5})}^{9}$ ⓑ ${(4^{4})}^{7}$ ⓒ ${(y^{3})}^{6} {(y^{5})}^{4} .$

ⓐ


Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.

ⓑ


Use the Power Property.
Simplify.

ⓒ

	${(y^{3})}^{6} {(y^{5})}^{4}$
Use the Power Property.	$y^{18} \cdot y^{20}$
Add the exponents.	$y to the power 38$

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

	$open parenthesis 2 x close parenthesis cubed$
What does this mean?	$2 x \cdot 2 x \cdot 2 x$
We group the like factors together.	$2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$
How many factors of 2 and of $x$	$2 cubed times x cubed$

Notice that each factor was raised to the power and $open parenthesis 2 x close parenthesis cubed$ is $2 cubed times x cubed.$

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Example 8

Simplify each expression: ⓐ $open parenthesis −3 m n close parenthesis cubed$ ⓑ $open parenthesis −4 a squared b close parenthesis to the power 0$ ⓒ $open parenthesis 6 k cubed close parenthesis to the power −2$ ⓓ $open parenthesis 5 x to the power −3 close parenthesis squared.$

ⓐ


Use Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$
Simplify.

ⓑ

	$open parenthesis −4 a squared b close parenthesis to the power 0$
Use Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$	${(−4)}^{0} {(a^{2})}^{0} {(b)}^{0}$
Simplify.	$1 \cdot 1 \cdot 1$
Multiply.	$1$

ⓒ

	$open parenthesis 6 k cubed close parenthesis to the power −2$
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.	$\frac{1}{36 k^{6}}$

ⓓ

	$open parenthesis 5 x to the power −3 close parenthesis squared$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$5 squared open parenthesis x to the power −3 close parenthesis squared$
Simplify.	$25 \cdot x^{−6}$
Rewrite $x to the power −6$ using, $a^{− n} = \frac{1}{a^{n}} .$	$25 \cdot \frac{1}{x^{6}}$
Simplify.	$\frac{25}{x^{6}}$

Now we will look at an example that will lead us to the Quotient to a Power Property.

	${(\frac{x}{y})}^{3}$
This means	$\frac{x}{y} \cdot \frac{x}{y} \cdot \frac{x}{y}$
Multiply the fractions.	$\frac{x \cdot x \cdot x}{y \cdot y \cdot y}$
Write with exponents.	$\frac{x^{3}}{y^{3}}$

Notice that the exponent applies to both the numerator and the denominator.

We see that ${(\frac{x}{y})}^{3}$ is $\frac{x^{3}}{y^{3}} .$

This leads to the Quotient to a Power Property for Exponents.

Example 9

Simplify each expression:

ⓐ ${(\frac{b}{3})}^{4}$ ⓑ ${(\frac{k}{j})}^{−3}$ ⓒ ${(\frac{2 x y^{2}}{z})}^{3}$ ⓓ ${(\frac{4 p^{−3}}{q^{2}})}^{2} .$

ⓐ

	$A mathematical expression showing the fraction b over 3, all raised to the power of 4.$
Use Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} .$	$A mathematical expression displaying b to the power of 4 divided by 3 to the power of 4, with both exponents in red. It can also be interpreted as the fraction b/3 raised to the power of 4.$
Simplify.	$A mathematical fraction showing 'b' raised to the power of 4, divided by 81.$

ⓑ


Raise the numerator and denominator to the power.	$A mathematical fraction showing k to the power of negative 3 over j to the power of negative 3, with the negative exponents in red.$
Use the definition of negative exponent.
Multiply.	$A mathematical expression displaying a fraction with 'j' cubed in the numerator and 'k' cubed in the denominator.$

ⓒ

	${(\frac{2 x y^{2}}{z})}^{3}$
Use Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} .$	$\frac{{(2 x y^{2})}^{3}}{z^{3}}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$\frac{8 x^{3} y^{6}}{z^{3}}$

ⓓ

	${(\frac{4 p^{−3}}{q^{2}})}^{2}$
Use Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} .$	$\frac{{(4 p^{−3})}^{2}}{{(q^{2})}^{2}}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$\frac{4^{2} {(p^{−3})}^{2}}{{(q^{2})}^{2}}$
Simplify using the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$\frac{16 p^{−6}}{q^{4}}$
Use the definition of negative exponent.	$\frac{16}{q^{4}} \cdot \frac{1}{p^{6}}$
Simplify.	$\frac{16}{p^{6} q^{4}}$

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

Note

Summary of Exponent Properties

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

Property	Description
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	${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$

Example 10

Simplify each expression by applying several properties:

ⓐ $open parenthesis 3 x squared y close parenthesis to the power 4 open parenthesis 2 x y squared close parenthesis cubed$ ⓑ $\frac{{(x^{3})}^{4} {(x^{−2})}^{5}}{{(x^{6})}^{5}}$ ⓒ ${(\frac{2 x y^{2}}{x^{3} y^{−2}})}^{2} {(\frac{12 x y^{3}}{x^{3} y^{−1}})}^{−1} .$

ⓐ

	$open parenthesis 3 x squared y close parenthesis to the power 4 open parenthesis 2 x y squared close parenthesis cubed$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$(3^{4} x^{8} y^{4}) (2^{3} x^{3} y^{6})$
Simplify.	$(81 x^{8} y^{4}) (8 x^{3} y^{6})$
Use the Commutative Property.	$81 \cdot 8 \cdot x^{8} \cdot x^{3} \cdot y^{4} \cdot y^{6}$
Multiply the constants and add the exponents.	$648 x^{11} y^{10}$

ⓑ

	$\frac{{(x^{3})}^{4} {(x^{−2})}^{5}}{{(x^{6})}^{5}}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$	$\frac{(x^{12}) (x^{−10})}{(x^{30})}$
Add the exponents in the numerator.	$\frac{x^{2}}{x^{30}}$
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}} .$	$\frac{1}{x^{28}}$

ⓒ

	${(\frac{2 x y^{2}}{x^{3} y^{−2}})}^{2} {(\frac{12 x y^{3}}{x^{3} y^{−1}})}^{−1}$
Simplify inside the parentheses first.	${(\frac{2 y^{4}}{x^{2}})}^{2} {(\frac{12 y^{4}}{x^{2}})}^{−1}$
Use the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} .$	$\frac{{(2 y^{4})}^{2}}{{(x^{2})}^{2}} \frac{{(12 y^{4})}^{−1}}{{(x^{2})}^{−1}}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m} .$	$\frac{4 y^{8}}{x^{4}} \cdot \frac{12^{−1} y^{−4}}{x^{−2}}$
Simplify.	$\frac{4 y^{4}}{12 x^{2}}$
Simplify.	$\frac{y^{4}}{3 x^{2}}$

Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite 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 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000	$4 times 1,000$	$4 times 10 cubed$
0.004	$4 \times \frac{1}{1,000}$	$4 \times \frac{1}{10^{3}}$	$4 \times 10^{−3}$

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 ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

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.

The figure shows two examples of converting from standard notation to scientific notation. In one example 4000 is converted to 4 times 10 to the power of 3. The decimal point in 4000 starts at the right and moves 3 places to the left to make the number 4. The 3 places moved make the exponent 3. In the other example, the number 0.004 is converted to 4 times 10 to the negative 3 power. The decimal point in 0.004 is moved 3 places to the right to make the number 4. The 3 places moved make the exponent negative 3.

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

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: $0.004 = 4 \times 10^{−3}$

Example 11

Write in scientific notation: ⓐ 37,000 ⓑ $0.0052.$

ⓐ

The original number, 37,000, is greater than 1 so we will have a positive power of 10.	37,000
Move the decimal point to get 3.7, 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: $\begin{matrix} 3.7 \times 10^{4} \\ 3.7 \times 10,000 \\ 37,000 \end{matrix}$

ⓑ

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.	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.
Write as a product with a power of 10.
$\begin{matrix} Check: & 5.2 \times 10^{−3} \\ 5.2 \times \frac{1}{10^{3}} \\ 5.2 \times \frac{1}{1000} \\ 5.2 \times 0.001 \\ 0.0052 \end{matrix}$

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.

The figure shows two examples of converting from scientific notation to standard notation. In one example 9.12 times 10 to the power of 4 is converted to 91200. The decimal point in 9.12 moves 4 places to the right to make the number 91200. In the other example, the number 9.12 times 10 to the power of -4 is converted to 0.000912. The decimal point in 9.12 is moved 4 places to the left to make the number 0.000912.

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 12

Convert to decimal form: ⓐ $6.2 times 10 cubed$ ⓑ $−8.9 \times 10^{−2} .$

ⓐ


Determine the exponent, n, on the factor 10.
	The exponent is 3.
Since the exponent is positive, move the decimal point 3 places to the right.
Add zeros as needed for placeholders.

ⓑ


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

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 13

Multiply or divide as indicated. Write answers in decimal form: ⓐ $(−4 \times 10^{5}) (2 \times 10^{−7})$ ⓑ $\frac{9 \times 10^{3}}{3 \times 10^{−2}} .$

ⓐ

	$(−4 \times 10^{5}) (2 \times 10^{−7})$
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.	$−0.08$

ⓑ

	$\frac{9 \times 10^{3}}{3 \times 10^{−2}}$
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.	$300,000$

Key Concepts

Exponential Notation

This is read a to the $m^{t h}$ power.
In the expression $a^{m}$ , the exponent m tells us how many times we use the base a as a factor.
Product Property for Exponents
If a is a real number and m and n are integers, then

$a^{m} \cdot a^{n} = a^{m + n}$

To multiply with like bases, add the exponents.
Quotient Property for Exponents
If $a$ is a real number, $a not equal to 0,$ and m and n are integers, then

$\frac{a^{m}}{a^{n}} = a^{m - n}, m > n and \frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}, n > m$
Zero Exponent
- If a is a non-zero number, then $a^{0} = 1 .$
- If a is a non-zero number, then a to the power of zero equals 1.
- Any non-zero number raised to the zero power is 1.
Negative Exponent
- If n is an integer and $a not equal to 0,$ then $a^{− n} = \frac{1}{a^{n}}$ or $\frac{1}{a^{− n}} = a^{n} .$
Quotient to a Negative Exponent Property
If $a, b$ are real numbers, $a not equal to 0, b not equal to 0$ and $n$ is an integer, then

${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$
Power Property for Exponents
If $a$ is a real number and $m, n$ are integers, then

${(a^{m})}^{n} = a^{m \cdot n}$

To raise a power to a power, multiply the exponents.
Product to a Power Property for Exponents
If a and b are real numbers and m is a whole number, then

${(a b)}^{m} = a^{m} b^{m}$

To raise a product to a power, raise each factor to that power.
Quotient to a Power Property for Exponents
If $a$ and are real numbers, $b not equal to 0,$ and $m$ is an integer, then

${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$

To raise a fraction to a power, raise the numerator and denominator to that power.

Summary of Exponent Properties
If a and b are real numbers, and m and n are integers, then

Property	Description
Product Property	$a^{m} \cdot a^{n} = a^{m + n}$
Power Property	${(a^{m})}^{n} = a^{m \cdot n}$
Product to a Power	${(a b)}^{n} = a^{n} b^{n}$
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	${(\frac{a}{b})}^{− n} = {(\frac{b}{a})}^{n}$

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

$a \times 10^{n} where 1 \leq a < 10 and n is an integer.$
How to convert a decimal to scientific notation.
1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
2. Count the number of decimal places, $n,$ that the decimal point was moved.
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 $10 to the power n.$
  - between 0 and 1, the power of 10 will be $10 to the power − n.$
4. Check.
How to convert scientific notation to decimal form.
1. Determine the exponent, $n,$ on the factor 10.
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 $| n |$ places to the left.
3. Check.

Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

ⓐ $d^{3} \cdot d^{6}$ ⓑ $4^{5 x} \cdot 4^{9 x}$ ⓒ $2 y times 4 y cubed$ ⓓ $w \cdot w^{2} \cdot w^{3}$

ⓐ $d to the power 9$ ⓑ $4 to the power 14 x$ ⓒ $8 y to the power 4$ ⓓ $w to the power 6$

ⓐ $x^{4} \cdot x^{2}$ ⓑ $8^{9 x} \cdot 8^{3}$ ⓒ $3 z^{25} \cdot 5 z^{8}$ ⓓ $y \cdot y^{3} \cdot y^{5}$

ⓐ $n^{19} \cdot n^{12}$ ⓑ $3^{x} \cdot 3^{6}$ ⓒ $7 w^{5} \cdot 8 w^{}$ ⓓ $a^{4} \cdot a^{3} \cdot a^{9}$

ⓐ $n to the power 31$ ⓑ $3^{x + 6}$ ⓒ $56 w to the power 6$
ⓓ $a to the power 16$

ⓐ $q^{27} \cdot q^{15}$ ⓑ $5^{x} \cdot 5^{4 x}$ ⓒ $9 u^{41} \cdot 7 u^{53}$
ⓓ $c^{5} \cdot c^{11} \cdot c^{2}$

$m^{x} \cdot m^{3}$

$m^{x + 3}$

$n^{y} \cdot n^{2}$

$y^{a} \cdot y^{b}$

$y^{a + b}$

$x^{p} \cdot x^{q}$

ⓐ $\frac{x^{18}}{x^{3}}$ ⓑ $\frac{5^{12}}{5^{3}}$ ⓒ $\frac{q^{18}}{q^{36}}$ ⓓ $\frac{10^{2}}{10^{3}}$

ⓐ $x to the power 15$ ⓑ $5 to the power 9$ ⓒ $\frac{1}{q^{18}}$ ⓓ $\frac{1}{10}$

ⓐ $\frac{y^{20}}{y^{10}}$ ⓑ $\frac{7^{16}}{7^{2}}$ ⓒ $\frac{t^{10}}{t^{40}}$ ⓓ $\frac{8^{3}}{8^{5}}$

ⓐ $\frac{p^{21}}{p^{7}}$ ⓑ $\frac{4^{16}}{4^{4}}$ ⓒ $\frac{b}{b^{9}}$ ⓓ $\frac{4}{4^{6}}$

ⓐ $p to the power 14$ ⓑ $4 to the power 12$ ⓒ $\frac{1}{b^{8}}$ ⓓ $\frac{1}{4^{5}}$

ⓐ $\frac{u^{24}}{u^{3}}$ ⓑ $\frac{9^{15}}{9^{5}}$ ⓒ $\frac{x}{x^{7}}$ ⓓ $\frac{10}{10^{3}}$

ⓐ $20 to the power 0$ ⓑ $b to the power 0$

ⓐ 1 ⓑ 1

ⓐ $13 to the power 0$ ⓑ $k to the power 0$

ⓐ $− 27 to the power 0$ ⓑ $− open parenthesis 27 to the power 0 close parenthesis$

ⓐ $−1$ ⓑ $−1$

ⓐ $− 15 to the power 0$ ⓑ $− open parenthesis 15 to the power 0 close parenthesis$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

ⓐ $a to the power −2$ ⓑ $10 to the power −3$ ⓒ $\frac{1}{c^{−5}}$ ⓓ $\frac{1}{3^{−2}}$

ⓐ $\frac{1}{a^{2}}$ ⓑ $\frac{1}{1000}$ ⓒ $c to the power 5$ ⓓ $9$

ⓐ $b to the power −4$ ⓑ $10 to the power −2$ ⓒ $\frac{1}{b^{−3}}$ ⓓ $\frac{1}{5^{−2}}$

ⓐ $r to the power −3$ ⓑ $10 to the power −5$ ⓒ $\frac{1}{q^{−10}}$ ⓓ $\frac{1}{10^{−3}}$

ⓐ $\frac{1}{r^{3}}$ ⓑ $\frac{1}{100,000}$ ⓒ $q to the power 10$
ⓓ $1,000$

ⓐ $s to the power −8$ ⓑ $10 to the power −2$ ⓒ $\frac{1}{t^{−9}}$ ⓓ $\frac{1}{10^{−4}}$

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

ⓐ $\frac{64}{25}$ ⓑ $\frac{a^{2}}{b^{2}}$

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

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

ⓐ $\frac{729}{64}$ ⓑ $- \frac{v^{5}}{u^{5}}$

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

ⓐ $open parenthesis −5 close parenthesis to the power −2$ ⓑ $− 5 to the power −2$ ⓒ ${(- \frac{1}{5})}^{−2}$ ⓓ $− {(\frac{1}{5})}^{−2}$

ⓐ $\frac{1}{25}$ ⓑ $- \frac{1}{25}$ ⓒ $25$ ⓓ $−25$

ⓐ $− 5 to the power −3$ ⓑ ${(- \frac{1}{5})}^{−3}$ ⓒ $− {(\frac{1}{5})}^{−3}$ ⓓ $open parenthesis −5 close parenthesis to the power −3$

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

ⓐ $\frac{3}{5}$ ⓑ $\frac{1}{15}$

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

In the following exercises, simplify each expression using the Product Property.

ⓐ $b^{4} b^{−8}$ ⓑ $(w^{4} x^{−5}) (w^{−2} x^{−4})$ ⓒ $(−6 c^{−3} d^{9}) (2 c^{4} d^{−5})$

ⓐ $\frac{1}{b^{4}}$ ⓑ $\frac{w^{2}}{x^{9}}$ ⓒ $−12 c d to the power 4$

ⓐ $s^{3} \cdot s^{−7}$ ⓑ $(m^{3} n^{−3}) (m^{−5} n^{−1})$ ⓒ $(−2 j^{−5} k^{8}) (7 j^{2} k^{−3})$

ⓐ $a^{3} \cdot a^{−3}$ ⓑ $(u v^{−2}) (u^{−5} v^{−3})$ ⓒ $(−4 r^{−2} s^{−8}) (9 r^{4} s^{3})$

ⓐ 1 ⓑ $\frac{1}{u^{4} v^{5}}$ ⓒ $\frac{- 36 r^{2}}{s^{5}}$

ⓐ $y^{5} \cdot y^{−5}$ ⓑ $(p q^{−4}) (p^{−6} q^{−3})$ ⓒ $(−5 m^{4} n^{6}) (8 m^{−5} n^{−3})$

$p^{5} \cdot p^{−2} \cdot p^{−4}$

$\frac{1}{p}$

$x^{4} \cdot x^{−2} \cdot x^{−3}$

In the following exercises, simplify each expression using the Power Property.

ⓐ $open parenthesis m to the power 4 close parenthesis squared$ ⓑ $open parenthesis 10 cubed close parenthesis to the power 6$ ⓒ $open parenthesis x cubed close parenthesis to the power −4$

ⓐ $m to the power 8$ ⓑ $10 to the power 18$ ⓒ $\frac{1}{x^{12}}$

ⓐ $open parenthesis b squared close parenthesis to the power 7$ ⓑ $open parenthesis 3 to the power 8 close parenthesis squared$ ⓒ $open parenthesis k squared close parenthesis to the power −5$

ⓐ ${(y^{3})}^{x}$ ⓑ ${(5^{x})}^{y}$ ⓒ ${(q^{6})}^{−8}$

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ $open parenthesis −3 x y close parenthesis squared$ ⓑ $open parenthesis 6 a close parenthesis to the power 0$ ⓒ $open parenthesis 5 x squared close parenthesis to the power −2$ ⓓ $open parenthesis −4 y to the power −3 close parenthesis squared$

ⓐ $open parenthesis −4 a b close parenthesis squared$ ⓑ $open parenthesis 5 x close parenthesis to the power 0$ ⓒ $open parenthesis 4 y cubed close parenthesis to the power −3$ ⓓ $open parenthesis −7 y to the power −3 close parenthesis squared$

ⓐ $open parenthesis −5 a b close parenthesis cubed$ ⓑ $open parenthesis −4 p q close parenthesis to the power 0$ ⓒ $open parenthesis −6 x cubed close parenthesis to the power −2$ ⓓ $open parenthesis 3 y to the power −4 close parenthesis squared$

ⓐ $open parenthesis −3 x y z close parenthesis to the power 4$ ⓑ $open parenthesis −7 m n close parenthesis to the power 0$ ⓒ $open parenthesis −3 x cubed close parenthesis to the power −2$
ⓓ $open parenthesis 2 y to the power −5 close parenthesis squared$

In the following exercises, simplify each expression using the Quotient to a Power Property.

ⓐ ${(\frac{x}{3})}^{4}$ ⓑ ${(\frac{a}{b})}^{−5}$ ⓒ ${(\frac{2 x^{2} y^{3}}{z^{2}})}^{2}$ ⓓ ${(\frac{x^{3} y}{z^{4}})}^{2}$

ⓐ ${(\frac{a}{3 b})}^{4}$ ⓑ ${(\frac{5}{4 m})}^{−2}$ ⓒ ${(\frac{3 a^{−2} b^{3}}{c^{2}})}^{−2}$ ⓓ ${(\frac{p^{−1} q^{4}}{r^{−4}})}^{2}$

ⓐ ${(\frac{x}{2 y})}^{3}$ ⓑ ${(\frac{10}{3 q})}^{−4}$ ⓒ ${(\frac{2 x^{3} y^{4}}{3 z^{2}})}^{5}$ ⓓ ${(\frac{5 a^{3} b^{−1}}{2 c^{4}})}^{−3}$

In the following exercises, simplify each expression by applying several properties.

ⓐ $open parenthesis 5 t squared close parenthesis cubed open parenthesis 3 t close parenthesis squared$ ⓑ $\frac{{(t^{2})}^{5} {(t^{−4})}^{2}}{{(t^{3})}^{7}}$ ⓒ ${(\frac{2 x y^{2}}{x^{3} y^{−2}})}^{2} {(\frac{12 x y^{3}}{x^{3} y^{−1}})}^{−1}$

ⓐ ${(10 k^{4})}^{3} {(5 k^{6})}^{2}$ ⓑ $\frac{{(q^{3})}^{6} {(q^{−2})}^{3}}{{(q^{4})}^{8}}$

ⓐ ${(m^{2} n)}^{2} {(2 m n^{5})}^{4}$ ⓑ $\frac{{(−2 p^{−2})}^{4} {(3 p^{4})}^{2}}{{(−6 p^{3})}^{2}}$

ⓐ $16 m^{8} n^{22}$ ⓑ $\frac{4}{p^{6}}$

ⓐ ${(3 p q^{4})}^{2} {(6 p^{6} q)}^{2}$ ⓑ $\frac{{(−2 k^{−3})}^{2} {(6 k^{2})}^{4}}{{(9 k^{4})}^{2}}$

Mixed Practice

In the following exercises, simplify each expression.

ⓐ $7 n to the power −1$ ⓑ $open parenthesis 7 n close parenthesis to the power −1$ ⓒ $open parenthesis −7 n close parenthesis to the power −1$

ⓐ $6 r to the power −1$ ⓑ $open parenthesis 6 r close parenthesis to the power −1$ ⓒ $open parenthesis −6 r close parenthesis to the power −1$

${(x^{2})}^{4} \cdot {(x^{3})}^{2}$

$x to the power 14$

${(y^{4})}^{3} \cdot {(y^{5})}^{2}$

${(a^{2})}^{6} \cdot {(a^{3})}^{8}$

$a to the power 36$

${(b^{7})}^{5} \cdot {(b^{2})}^{6}$

$open parenthesis 2 m to the power 6 close parenthesis cubed$

$8 m to the power 18$

$open parenthesis 3 y squared close parenthesis to the power 4$

$open parenthesis 10 x squared y close parenthesis cubed$

$1,000 x to the power 6 y cubed$

${(2 m n^{4})}^{5}$

$open parenthesis −2 a cubed b squared close parenthesis to the power 4$

$16 a^{12} b^{8}$

$open parenthesis −10 u squared v to the power 4 close parenthesis cubed$

${(\frac{2}{3} x^{2} y)}^{3}$

$\frac{8}{27} x^{6} y^{3}$

${(\frac{7}{9} p q^{4})}^{2}$

$open parenthesis 8 a cubed close parenthesis squared open parenthesis 2 a close parenthesis to the power 4$

$1,024 a to the power 10$

$open parenthesis 5 r squared close parenthesis cubed open parenthesis 3 r close parenthesis squared$

${(10 p^{4})}^{3} {(5 p^{6})}^{2}$

$25,000 p to the power 24$

${(4 x^{3})}^{3} {(2 x^{5})}^{4}$

${(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}$

$x^{18} y^{18}$

${(\frac{1}{3} m^{3} n^{2})}^{4} {(9 m^{8} n^{3})}^{2}$

${(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}$

$144 m^{8} n^{22}$

${(2 p q^{4})}^{3} {(5 p^{6} q)}^{2}$

ⓐ $open parenthesis 3 x close parenthesis squared open parenthesis 5 x close parenthesis$ ⓑ $open parenthesis 2 y close parenthesis cubed open parenthesis 6 y close parenthesis$

ⓐ $45 x cubed$ ⓑ $48 y to the power 4$

ⓐ ${(\frac{1}{2} y^{2})}^{3} {(\frac{2}{3} y)}^{2}$ ⓑ ${(\frac{1}{2} j^{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

ⓐ ${(2 r^{−2})}^{3} {(4^{−1} r)}^{2}$ ⓑ ${(3 x^{−3})}^{3} {(3^{−1} x^{5})}^{4}$

ⓐ $\frac{1}{2 r^{4}}$ ⓑ $\frac{1}{3} x^{11}$

${(\frac{k^{−2} k^{8}}{k^{3}})}^{2}$

${(\frac{j^{−2} j^{5}}{j^{4}})}^{3}$

$\frac{1}{j^{3}}$

$\frac{{(−4 m^{−3})}^{2} {(5 m^{4})}^{3}}{{(−10 m^{6})}^{3}}$

$\frac{{(−10 n^{−2})}^{3} {(4 n^{5})}^{2}}{{(2 n^{8})}^{2}}$

$- \frac{4000}{n^{12}}$

Use Scientific Notation

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

ⓐ 57,000 ⓑ 0.026

ⓐ 340,000 ⓑ 0.041

ⓐ $3.4 \times 10^{5}$ ⓑ $4.1 \times 10^{−2}$

ⓐ 8,750,000 ⓑ 0.00000871

ⓐ 1,290,000 ⓑ 0.00000103

ⓐ $1.29 \times 10^{6}$
ⓑ $1.03 \times 10^{−6}$

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

ⓐ $5.2 times 10 squared$ ⓑ $2.5 \times 10^{−2}$

ⓐ $−8.3 times 10 squared$ ⓑ $3.8 \times 10^{−2}$

ⓐ $−830$ ⓑ 0.038

ⓐ $7.5 \times 10^{6}$ ⓑ $−4.13 \times 10^{−5}$

ⓐ $1.6 \times 10^{10}$ ⓑ $8.43 \times 10^{−6}$

ⓐ 16,000,000,000
ⓑ 0.00000843

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

ⓐ $(3 \times 10^{−5}) (3 \times 10^{9})$ ⓑ $\frac{7 \times 10^{−3}}{1 \times 10^{−7}}$

ⓐ $(2 \times 10^{2}) (1 \times 10^{−4})$ ⓑ $\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

ⓐ 0.02 ⓑ 500,000,000

ⓐ $(7.1 \times 10^{−2}) (2.4 \times 10^{−4})$ ⓑ $\frac{6 \times 10^{4}}{3 \times 10^{−2}}$

ⓐ $(3.5 \times 10^{−4}) (1.6 \times 10^{−2})$ ⓑ $\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

ⓐ 0.0000056 ⓑ 20,000,000

Writing Exercises

Use the Product Property for Exponents to explain why $x \cdot x = x^{2} .$

Jennifer thinks the quotient $\frac{a^{24}}{a^{6}}$ simplifies to $a to the power 4.$ What is wrong with her reasoning?

Answers will vary.

Explain why $− 5 cubed equals open parenthesis −5 close parenthesis cubed$ but $− 5^{4} \neq {(−5)}^{4} .$

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

Answers will vary.

Self Check

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

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions using the properties for exponents.”, “use the definition of a negative exponent”, and “use scientific notation”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?