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.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This is read a to the $m^{th}$ power.

In the expression $a^m,$ the exponent m tells us how many times we use the base a as a factor.

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.

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What does this mean?	$$
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Notice that 5 is the sum of the exponents, 2 and 3. We see $x^2·x^3$ is $x^{2+3}$ or $x^5.$

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

Example 5.12

Simplify each expression: ⓐ $y^5·y^6$ ⓑ $2^x·2^{3x}$ ⓒ $2a^7·3a.$

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Solution

ⓐ

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

ⓑ

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

ⓒ

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

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. $\frac{\cancel{x}}{\cancel{x}}=1$. This leads to the Quotient Property for Exponents.

Example 5.13

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

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Solution

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

ⓐ

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

ⓑ

Since $10>2,$ there are more factors of $3$ in the numerator.$$
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 $12>8,$ there are more factors of $b$ in the denominator.
Use Quotient Property, $\frac{a^m}{a^n}=\frac{1}{a^{n-m}}.$
Simplify.

ⓓ

Since $5>3,$ there are more factors of $3$ in the denominator.$$
Use Quotient Property, $\frac{a^m}{a^n}=\frac{1}{a^{n-m}}.$
Simplify.
Simplify.

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 $x\left(x\ne 0\right)$ 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 $a\ne 0:$

We see $\frac{a^m}{a^m}$ simplifies to $a^0$ 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 $x^{\mathrm{-3}}.$

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

This implies that $x^{\mathrm{-3}}=\frac{1}{x^3}$ and it leads us to the definition of a negative exponent. If n is an integer and $a\ne 0,$ then $a^{\text{−}n}=\frac{1}{a^n}.$

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.

$\begin{array}{cccccc}& & & & & \hfill \frac{1}{a^{\text{−}n}}\hfill \\ \\ \\ \text{Use the definition of a negative exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & & & \hfill \frac{1}{\frac{1}{a^n}}\hfill \\ \\ \\ \text{Simplify the complex fraction.}\hfill & & & & & \hfill 1·\frac{a^n}{1}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & & & \hfill a^n\hfill \end{array}$

This implies $\frac{1}{a^{\text{−}n}}=a^n$ 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 $x^{\mathrm{-3}},$ 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.

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.

$\begin{array}{cccccc}& & & & & \hfill {\left(\frac{3}{4}\right)}^{\mathrm{-2}}\hfill \\ \\ \\ \text{Use the definition of a negative exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & & & \hfill \frac{1}{{\left(\frac{3}{4}\right)}^2}\hfill \\ \\ \\ \text{Simplify the denominator.}\hfill & & & & & \hfill \frac{1}{\frac{9}{16}}\hfill \\ \\ \\ \text{Simplify the complex fraction.}\hfill & & & & & \hfill \frac{16}{9}\hfill \\ \\ \\ \text{But we know that}\frac{16}{9}\text{is}{\left(\frac{4}{3}\right)}^2.\hfill & & & & & \\ \text{This tells us that}\hfill & & & & & \hfill {\left(\frac{3}{4}\right)}^{\mathrm{-2}}={\left(\frac{4}{3}\right)}^2\hfill \end{array}$

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.

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

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

$\begin{array}{cccccc}& & & & & \hfill {(x^2)}^3\hfill \\ \text{What does this mean?}\hfill & & & & & \hfill x^2·x^2·x^2\hfill \end{array}$

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 $x^{2·3}$ or $x^6.$

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

Example 5.18

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

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Solution

ⓐ

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

ⓑ

Use the Power Property.
Simplify.

ⓒ
$\begin{array}{cccccc}& & & & & \hfill {(y^3)}^6{(y^5)}^4\hfill \\ \text{Use the Power Property.}\hfill & & & & & \hfill y^{18}·y^{20}\hfill \\ \text{Add the exponents.}\hfill & & & & & \hfill y^{38}\hfill \end{array}$

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

$\begin{array}{cccccc}& & & & & \hfill {\left(2x\right)}^3\hfill \\ \text{What does this mean?}\hfill & & & & & \hfill 2x·2x·2x\hfill \\ \text{We group the like factors together.}\hfill & & & & & \hfill 2·2·2·x·x·x\hfill \\ \text{How many factors of 2 and of}x\hfill & & & & & \hfill 2^3·x^3\hfill \end{array}$

Notice that each factor was raised to the power and ${\left(2x\right)}^3$ is $2^3·x^3.$

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

Example 5.19

Simplify each expression: ⓐ ${(\mathrm{-3}mn)}^3$ ⓑ ${\left(\mathrm{-4}{a}^{2}b\right)}^0$ ⓒ ${\left(6k^3\right)}^{\mathrm{-2}}$ ⓓ ${\left(5x^{\mathrm{-3}}\right)}^2.$

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Solution

ⓐ

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

ⓑ
$\begin{array}{cccccc}& & & & & \hfill {\left(\mathrm{-4}{a}^{2}b\right)}^0\hfill \\ \text{Use Power of a Product Property,}{(ab)}^m=a^m{b}^m.\hfill & & & & & \hfill {\left(\mathrm{-4}\right)}^0{\left(a^2\right)}^0{\left(b\right)}^0\hfill \\ \text{Simplify.}\hfill & & & & & \hfill 1·1·1\hfill \\ \text{Multiply.}\hfill & & & & & \hfill 1\hfill \end{array}$

ⓒ
$\begin{array}{cccccc}& & & & & \hfill {\left(6k^3\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Product to a Power Property,}{\left(ab\right)}^m=a^m{b}^m.\hfill & & & & & \hfill {\left(6\right)}^{\mathrm{-2}}{\left(k^3\right)}^{\mathrm{-2}}\hfill \\ \text{Use the Power Property,}{\left(a^m\right)}^n=a^{m·n}.\hfill & & & & & \hfill 6^{\mathrm{-2}}{k}^{\mathrm{-6}}\hfill \\ \text{Use the Definition of a negative exponent,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & & & \hfill \frac{1}{6^2}·\frac{1}{k^6}\hfill \\ \text{Simplify.}\hfill & & & & & \hfill \frac{1}{36k^6}\hfill \end{array}$

ⓓ
$\begin{array}{cccccc}& & & & & \hfill {\left(5x^{\mathrm{-3}}\right)}^2\hfill \\ \text{Use the Product to a Power Property,}{\left(ab\right)}^m=a^m{b}^m.\hfill & & & & & \hfill 5^2{\left(x^{\mathrm{-3}}\right)}^2\hfill \\ \text{Simplify.}\hfill & & & & & \hfill 25·x^{\mathrm{-6}}\hfill \\ \text{Rewrite}{x}^{\mathrm{-6}}\text{using,}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & & & \hfill 25·\frac{1}{x^6}\hfill \\ \text{Simplify.}\hfill & & & & & \hfill \frac{25}{x^6}\hfill \end{array}$