5.6: Power Rule For Exponents

This rule helps to simplify an exponential expression raised to a power. This rule is often confused with the product rule, so understanding this rule is important to successfully simplify exponential expressions.

Definition: The Power Rule For Exponents

For any real number \(a\) and any numbers \(m\) and \(n\), the power rule for exponents is the following:

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

Idea:

Given the expression

\(\begin{aligned} &(2^2 )^3 && \text{Use the exponent definition to expand the expression inside the parentheses.} \\ &(2 \cdot 2)^3 && \text{Now use the exponent definition to expand according to the exponent outside the parentheses.}\\ &(2 \cdot 2) \cdot (2 \cdot 2) \cdot (2 \cdot 2) = 2^6 && = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^{1+1+1+1+1+1 }= 2^{6} \text{ (Product Rule of Exponents) }\end{aligned}\)

Hence, \((2^2 ) ^3 = 2^{2\cdot 3 }= 2^6\)

Example 5.6.1

Simplify the following expression using the power rule for exponents.

\((−3^4 )^3\)

Solution

\((−3)^{4\cdot 3 }= (−3)^{12}\)

Example 5.6.2

Simplify the following expression using the power rule for exponents.

\((−3^4 )^3\)

Solution

\((5y)^{3\cdot 7 }= (5y)^{21}\)

Example 5.6.3

Simplify the following expression using the power rule for exponents.

\(((−y)^5 )^2\)

Solution

\((−y)^{5\cdot 2 }= (−y)^{10 }= y^{10}\)

Example 5.6.4

Simplify the following expression using the power rule for exponents.

\((x^{−2 })^3\)

Solution

\(x^{−2\cdot 3 }= x^{−6 }= \dfrac{1 }{x^6}\)

Hint: Parentheses in the problem is a strong indicator of simplifying using the power rule for exponents.

Exercise 5.6.1

Simplify the expression using the power rule for exponents.

1. \((x^3 )^5\)
2. \(((−y)^3 )^7\)
3. \(((−6y)^8 ) ^{−3}\)
4. \((x^{−2 }) ^{−3}\)
5. \((r^4 )^5\)
6. \((−p^7 )^7\)
7. \(((3k)^{−3 })^5\)