5.7: The power of a product rule for exponents
The power of a product rule for exponents will deal with expressions where a product of bases is raised to some power.
For any real number a and b and any number n, the power of a product rule for exponents is the following:
\((a \cdot b)^n =a^n \cdot b^n\)
Simplify the following expression
\((a \cdot b)^3\)
Solution
\(\begin{aligned} &(a \cdot b) 3&& \text{Given} \\ &a \cdot b \cdot a \cdot b \cdot a \cdot b &&\text{Expand using exponent 3} \\ &a \cdot a \cdot a \cdot b \cdot b \cdot b &&\text{Reorder product using the commutative property.} \\ &a^3b^3 && \text{Simplify to single base.} \end{aligned}\)
Simplifying expression using the power of a product rule for exponents.
\((2x^3y^2 )^2 \)
Solution
\(\begin{aligned} &((2x^3y^2 )^2 ) &&\text{Given} \\ &= 2^2 \cdot x^{3\cdot 2 } \cdot y^{2\cdot 2 } &&\text{Power of a product rule applied} \\ &= 4x ^6y^4 &&\text{Simplify by multiplying exponents} \end{aligned}\)
Simplifying expression using the power of a product rule for exponents.
\((2x^5 \cdot 3y)^3\)
Solution
\(\begin{aligned} &(2x^5 \cdot 3y)^3 &&\text{Given}\\ &= (6x^5 y)^3 &&\text{If possible, simplify inside the parentheses.} \\ &= 6^3 \cdot x^{5\cdot 3 } \cdot y^3 &&\text{Power of a product rule for exponents applied.} \\ &216x^{15}y^3 &&\text{Simplify by multiplying as needed.} \end{aligned}\)
Simplifying expression using the power of a product rule for exponents.
\((3ab^4 )^{−2}\)
Solution
\(\begin{aligned} &(3ab^4 )^{−2 } && \text{Given} \\ &= \dfrac{1 }{(3ab^4)^2 } &&\text{Negative exponent rule applied} \\ &= \dfrac{1 }{3^2 \cdot a^3 \cdot b^{4\cdot 2}}&&\text{Power of a product rule applied.} \\ &\dfrac{1 }{9a^3b^8 } &&\text{Simplify by multiplying as needed.} \end{aligned}\)
Simplify the expression using the power of a product rule for exponents.
- \((xy^2 ) ^3\)
- \((2xy^2z ^3 ) ^7\)
- \((x^ 2 ) ^{−3}\)
- \((x ^2y) ^{−3}\)
- \((2r ^4 )^ 5\)
- \((−2p^ 7 )^ 7\)
- \((3k \cdot 2j^2 )^5\)