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.
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, \((2^2 ) ^3 = 2^{2\cdot 3 }= 2^6\)
Simplify the following expression using the power rule for exponents.
\((−3^4 )^3\)
Solution
\((−3)^{4\cdot 3 }= (−3)^{12}\)
Simplify the following expression using the power rule for exponents.
\((−3^4 )^3\)
Solution
\((5y)^{3\cdot 7 }= (5y)^{21}\)
Simplify the following expression using the power rule for exponents.
\(((−y)^5 )^2\)
Solution
\((−y)^{5\cdot 2 }= (−y)^{10 }= y^{10}\)
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.
Simplify the expression using the power rule for exponents.
- \((x^3 )^5\)
- \(((−y)^3 )^7\)
- \(((−6y)^8 ) ^{−3}\)
- \((x^{−2 }) ^{−3}\)
- \((r^4 )^5\)
- \((−p^7 )^7\)
- \(((3k)^{−3 })^5\)