1.4.2: The Power Rules for Exponents

1.4.3 Learning Objectives

• Use the Power Rule for Powers to simplify expressions involving exponents
• Use the Power Rule for Products to simplify expressions involving exponents
• Use the Power Rule for Quotients to simplify expressions involving exponents

The Power Rule for Powers

The following examples suggest a rule for raising a power to a power:

\((a^2)^3 = a^2 \cdot a^2 \cdot a^2\)
Using the product rule we get:
\((a^2)^3 = a^{2+2+2}\)
\((a^2)^3 = a^{3 \cdot 2}\)
\((a^2)^3 = a^6\)

\((x^9)^4 = x^9 \cdot x^9 \cdot x^9 \cdot x^9\)
\((x^9)^4 = x^{9+9+9+9}\)
\((x^9)^4 = x^{4 \cdot 9}\)
\((x^9)^4 = x^{36}\)

Power Rule for Powers

If \(x\) is a real number and \(n\) and \(m\) are natural numbers,
\((x^n)^m = x^{n \cdot m}\)

To raise a power to a power, multiply the exponents.

 

Example 1

Simplify each expression using the power rule for powers. All exponents are natural numbers.

1. \((7^3)^4 = 7^{3 \cdot 4} = 7^{12}\)
2. \((y^5)^3 = y^{5 \cdot 3} = y^{15}\)
3. \((d^{20})^6 = d^{20 \cdot 6} = d^{120}\)

 

Try It Now 1

Simplify the expression using the power rule for powers.

\((8^5)^4\)

Answer

\(8^{20}\)

Try It Now 2

Simplify the expression using the power rule for powers.

\((y^7)^7\)

Answer

\(y^{49}\)

The Power Rule for Products

The following examples suggest a rule for raising a product to a power:

\(
\begin{aligned}
&(a b)^{3}=a b \cdot a b \cdot a b \text { Use the commutative property of multiplication. }\\
&\begin{array}{l}
=a a a b b b \\
=a^{3} b^{3}
\end{array}
\end{aligned}
\)

\(
\begin{aligned}
(x y)^{5} &=x y \cdot x y \cdot x y \cdot x y \cdot x y \\
&=x x x x x \cdot \text { yyyyy } \\
&=x^{5} y^{5}
\end{aligned}
\)

\(
\begin{aligned}
(4 x y \mathrm{z})^{2} &=4 x y z \cdot 4 x y z \\
&=4 \cdot 4 \cdot x x \cdot y y \cdot z z \\
&=16 x^{2} y^{2} z^{2}
\end{aligned}
\)

Power Rule for Products

If \(x\) and \(y\) are real numbers are \(n\) is a natural number,
\((xy)^n = x^ny^n\)

To raise a product to a power, apply the exponent rule to each and every factor

 

Example 2

Make use of either the power rule for products or power rule for powers to simplify each expression.

1. \((2b)^7 = 2^7b^7\)
2. \((axy)^4 = a^4x^4y^4\)

3. \((3ab)^2 = 3^2a^2b^2 = 9a^2b^2\)

   Don't forget to apply the exponent to the 3!

Example 3

Make use of both the power rule for products and power rule for powers to simplify each expression.

1. \((ab^3)^2 = a^2(b^3)^2 = a^2b^6\)

   We used two rules here. First, the power rule for products. Second, the power rule for powers.

2. \((7a^4b^2c^8)^2 = 7^2(a^4)^2(b^2)^2(c^8)^2 = 49a^8b^4c^{16}\)
3. \([2(x+1)^4]^6 = 2^6(x+1)^{24} = 64(x+1)^{24}\)

 

Example 4

If \(6a^3c^7 \not = 0\), then \((6a^3c^7)^0 = 1\). Recall that \(x^0 = 1\) for \(x \not = 0\)

 

The Power Rule for Quotients

The following example suggests a rule for raising a quotient to a power.

\((\dfrac{a}{b})^3 = \dfrac{a}{b} \cdot \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a \cdot a \cdot a}{b \cdot b \cdot b} = \dfrac{a^3}{b^3}\)

Power Rule for Quotients

If \(x\) and \(y\) are real numbers and \(n\) is a natural number,

\((\dfrac{x}{y})^n = \dfrac{x^n}{y^n}, y \not = 0\)

To raise a quotient to a power, distribute the exponent to both the numerator and denominator.

 

Example 5

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. All exponents are natural numbers.

1. \((\dfrac{x}{y})^6 = \dfrac{x^6}{y^6}\)
2. \((\dfrac{2x}{b})^4 = \dfrac{(2x)^4}{b^4} = \dfrac{2^4x^4}{b^4} = \dfrac{16x^4}{b^4}\)
3. \((\dfrac{a^3}{b^5})^7 = \dfrac{(a^3)^7}{(b^5)^7} = \dfrac{a^21}{b^35}\)