1Cademy - Simplifying $$(m^3n^9)^{\frac{1}{3}}$$ and $$(p^4q^8)^{\frac{1}{4}}$$ Using the Product to a Power Property with Rational Exponents

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Product to a Power Property for Exponents

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Simplifying $(m^3n^9)^{\frac{1}{3}}$ and $(p^4q^8)^{\frac{1}{4}}$ Using the Product to a Power Property with Rational Exponents

Apply the Product to a Power Property and the Power Property together to simplify two expressions in which a product of variable powers is raised to a rational exponent of the form $\frac{1}{n}$ . First distribute the rational exponent to each factor using $(ab)^m = a^m b^m$ , then multiply the exponents in each power-of-a-power using $(a^m)^n = a^{m \cdot n}$ .

ⓐ $(m^3 n^9)^{\frac{1}{3}} = mn^3$ : The base is the product $m^3 n^9$ , and the outer exponent is $\frac{1}{3}$ . Use the Product to a Power Property to distribute the exponent to each factor: $(m^3)^{\frac{1}{3}} (n^9)^{\frac{1}{3}}$ . Now apply the Power Property to each factor by multiplying the exponents: $m^{3 \cdot \frac{1}{3}} \cdot n^{9 \cdot \frac{1}{3}} = m^1 \cdot n^3 = mn^3$ .

ⓑ $(p^4 q^8)^{\frac{1}{4}} = pq^2$ : The base is the product $p^4 q^8$ , and the outer exponent is $\frac{1}{4}$ . Distribute the exponent: $(p^4)^{\frac{1}{4}} (q^8)^{\frac{1}{4}}$ . Multiply the exponents: $p^{4 \cdot \frac{1}{4}} \cdot q^{8 \cdot \frac{1}{4}} = p^1 \cdot q^2 = pq^2$ .

In both parts, the same two-step procedure applies: first distribute the rational exponent across the product, then simplify each power-of-a-power by multiplying the integer exponent inside by the fractional exponent outside. When the inner exponent is a multiple of the denominator in the outer exponent, the product simplifies to a whole number, yielding a clean integer exponent in the final answer.

Updated 2026-04-21

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OpenStax Elementary Algebra

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