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# Using Properties of Exponents

The properties of exponents are a set of tools that can be used to simplify expressions with exponents.
Rule

## Properties of Exponents

The properties of exponents allow expressions with exponents to be rewritten.

Rule

## Product of Powers Property

### Rule

$a^m\cdot a^n=a^{m+n}$
When powers with the same base are multiplied, they can be written as one power by adding the exponents. For example, $2^3 \cdot 2^2,$ can be expressed as follows using this rule. $2^3 \cdot 2^2 \Rightarrow 2^{3+2}=2^5$ This rule can be explained by writing the powers as repeated multiplication.
$2^3 \cdot 2^2$
$(2 \cdot 2 \cdot 2) \cdot (2 \cdot 2)$
$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$2^5$
This rule is valid for all values of $m$ and $n$ when $a\neq 0.$ For the rule to be true when $a=0$ it is necessary that $m, n> 0.$ Note that the rule might lead to non-real solutions when $a<0$ and at least one of the exponents is not an integer.
Rule

## Quotient of Powers Property

### Rule

$\dfrac{a^m}{a^n}=a^{m-n}$
When powers with the same base are divided, they can be written as one power where the exponent in the denominator is subtracted from the exponent in the numerator. For example, $3^6$ divided by $3^4$ can be rewritten using the rule as follows. $\dfrac{3^6}{3^4} \Rightarrow 3^{6-4} = 3^2$ This rule can be explained by writing the powers as a product.
$\dfrac{3^6}{3^4}$
$\dfrac{3\cdot 3\cdot 3\cdot3\cdot3\cdot3}{3\cdot3\cdot3\cdot3}$
$\dfrac{3\cdot 3\cdot \cancel{3}\cdot\cancel{3}\cdot\cancel{3}\cdot\cancel{3}}{\cancel{3}\cdot\cancel{3}\cdot\cancel{3}\cdot\cancel{3}}$
$3\cdot3$
$3^2$
This rule is valid for all values of $m,$ $n$ and when $a\neq 0.$ Note that the rule might lead to non-real solutions when $a<0$ and at least one of the exponents is not an integer.
Rule

## Power of a Power Property

### Rule

$\left(a^m\right)^n=a^{mn}$
When a power is raised to another exponent, the powers can be combined into one by multiplying them. For example $\left(5^2\right)^3$ can be rewritten with this rule as follows. $\left(5^2\right)^3 \Rightarrow 5^{2 \cdot 3} = 5^6$ This rule can be explained by writing the powers as products.
$\left(5^2\right)^3$
$a^3=a\cdot a\cdot a$
$5^2 \cdot 5^2 \cdot 5^2$
$5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$
$5^{6}$
This rule is valid for all values of $m$ and $n$ when $a\neq 0.$ For the rule to be true also for $a=0$ it is necessary that both exponents are greater than $0.$ Note that the rule might lead to non-real solutions when $a<0$ and at least one of the exponents is not an integer.
Rule

## Power of a Product Property

### Rule

$(ab)^m=a^m b^m$

When the base in an exponential expression is a product raised to some power, the term can be rewritten as a product of two powers. For example, $\left(2\cdot 5\right)^3$ can be rewritten using this rule as follows. $\left(2\cdot 5\right)^3 \Rightarrow 2^3 \cdot 5^3$ This rule can be explained by writing the expression with repeated multiplication.
$\left(2\cdot 5\right)^3$
$a^3=a\cdot a\cdot a$
$(2\cdot 5) \cdot (2\cdot 5) \cdot (2\cdot 5)$
$2\cdot 5 \cdot 2\cdot 5 \cdot 2\cdot 5$
$2\cdot 2 \cdot 2\cdot 5 \cdot 5\cdot 5$
$2^3\cdot 5^3$
This rule is valid for all values of $m,$ when $a\neq0$ and $b\neq 0.$ For the rule to be valid when either $a$ or $b$ equals $0$ it is necessary that $m>0.$ Note that the rule might lead to non-real solutions when at least one of the bases is less than $0$ and the exponent is not an integer.
Rule

## Power of a Quotient Property

### Rule

$\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$

When the base of a power is a fraction, the term can be rewritten as a fraction of two powers. The rule states that the exponent is applied to the numerator and the denominator. For example, $\left(\frac{6}{5}\right)^4$ can be rewritten using this rule as follows. $\left(\frac{6}{5}\right)^4 \Rightarrow \dfrac{6^4}{5^4}$ This rule can be explained by writing the power with repeated multiplication.
$\left(\dfrac{6}{5}\right)^4$
$\dfrac{6}{5} \cdot \dfrac{6}{5}\cdot\dfrac{6}{5}\cdot\dfrac{6}{5}$
$\dfrac{6\cdot 6\cdot6\cdot6}{5\cdot 5\cdot5\cdot5}$
$\dfrac{6^4}{5^4}$
For this rule to be valid it is necessary that $b\neq 0.$ Also, for the rule to be valid when $m\leq0$ it is necessary that $a\neq 0.$ Note that the rule might lead to non-real solutions when at least one of the bases is negative and $m$ is not an integer.
Rule

## Negative Exponent

### Rule

$a^{\text{-} n}=\dfrac{1}{a^n}$

When an exponent of a power is negative, the term can be rewritten as a fraction, with the base in the denominator. Note that the sign of the exponent is changed. For example, $5^{\text{-} 3}$ can be rewritten with this rule as follows. $\begin{gathered} 5^{\text{-} 3} \Rightarrow \dfrac{1}{5^3} \end{gathered}$ This rule can be explained by writing the exponent $\text{-}3$ as $4-7$ and applying the Quotient of Powers Property.
$5^{\text{-}3}$
$5^{4-7}$
$\dfrac{5^{4}}{5^{7}}$
$\dfrac{5\cdot5\cdot5\cdot5}{5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5}$
$\dfrac{\cancel{5}\cdot\cancel{5}\cdot\cancel{5}\cdot\cancel{5}}{5\cdot5\cdot5\cdot\cancel{5}\cdot\cancel{5}\cdot\cancel{5}\cdot\cancel{5}}$
$\dfrac{1}{5\cdot5\cdot5}$
$\dfrac{1}{5^3}$
This rule is valid for all values of $n$ and when $a\neq 0.$ Note that the rule might lead to non-real solutions when $a< 0$ and $n$ is not an integer.

Rule

## Zero Exponent

### Rule

$a^0=1$

Powers with an exponent of $0$ are equal to $1.$ This can be explained by writing an exponent of $0$ as a difference, for instance $2-2,$ and using the Quotient of Powers Property. Consider $4^0.$
$4^0$
Rewrite $0$ as $2-2$
$4^{2-2}$
$\dfrac{4^2}{4^2}$
$1$
This rule is valid for all values $a\neq 0.$
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Exercise

Simplify the following expression using properties of exponents. $\dfrac{12^{\text{-} 2}\cdot a^0\cdot b^4}{b^5}$

Show Solution
Solution
To simplify the exponents in an expression, it can be helpful to focus on either one power at a time or powers with the same base. Here, we will begin with $a^0.$ The Zero Exponent Property tells us that any base with an exponent of $0$ equals $1.$
$\dfrac{12^{\text{-} 2}\cdot a^0\cdot b^4}{b^5}$
$\dfrac{12^{\text{-} 2}\cdot 1\cdot b^4}{b^5}$
$\dfrac{12^{\text{-} 2}\cdot b^4}{b^5}$
Next, we can focus on the powers with base $b.$ Notice that there is one in the numerator and one in the denominator. We can use the Quotient of Powers Property to simplify these.
$\dfrac{12^{\text{-} 2}\cdot b^4}{b^5}$
$12^{\text{-} 2} \cdot \dfrac{b^4}{b^5}$
$12^{\text{-} 2}\cdot b^{4-5}$
$12^{\text{-} 2}\cdot b^{\text{-} 1}$
Lastly, we can rewrite the negative exponents using the Negative Exponent Property.
$12^{\text{-} 2}\cdot b^{\text{-} 1}$
$\dfrac{1}{12^ 2} \cdot \dfrac{1}{b^1}$
$\dfrac{1}{12^ 2} \cdot \dfrac{1}{b}$
$\dfrac{1}{12^ 2\cdot b}$
We have now completely simplified the expression. $\dfrac{1}{12^ 2\cdot b}$
Method

## Variable Exponents

The properties of exponents are not only valid for numerical exponents. They can be applied in the same way when exponents contain variables.
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Exercise

Simplify the following expression using the properties of exponents. $\dfrac{3^{2y}\cdot 3^{5y}}{3^y}$

Show Solution
Solution
To begin, we notice that each power in this expression has the same base. We will first focus on simplifying the numerator by applying the Product of Powers Property.
$\dfrac{3^{2y}\cdot 3^{5y}}{3^y}$
$\dfrac{3^{2y + 5y}}{3^y}$
$\dfrac{3^{7y}}{3^y}$
We can now simplify the fraction using the Quotient of Powers Property.
$\dfrac{3^{7y}}{3^y}$
$3^{7y-y}$
$3^{6y}$
The given expression simplifies completely to $3^{6y}.$