1Cademy - Simplifying $$\left(\frac{2xy^2}{z}\right)^3$$ and $$\left(\frac{3ab^3}{c^2}\right)^4$$ Using Exponent Properties

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Simplifying $\left(\frac{3}{7}\right)^2$ , $\left(\frac{b}{3}\right)^4$ , and $\left(\frac{k}{j}\right)^3$ Using the Quotient to a Power Property

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Simplifying $\left(\frac{2xy^2}{z}\right)^3$ and $\left(\frac{3ab^3}{c^2}\right)^4$ Using Exponent Properties

Apply multiple exponent properties—the Quotient to a Power Property, the Product to a Power Property, and the Power Property—to simplify rational expressions raised to an outer power.

ⓐ $\left(\frac{2xy^2}{z}\right)^3 = \frac{8x^3y^6}{z^3}$ : First, apply the Quotient to a Power Property to distribute the outer exponent $3$ to both the numerator and the denominator: $\frac{(2xy^2)^3}{z^3}$ . Next, use the Product to a Power Property in the numerator to distribute the $3$ to each factor: $\frac{2^3 x^3 (y^2)^3}{z^3}$ . Finally, evaluate $2^3 = 8$ and use the Power Property to multiply the exponents on $y$ ( $2 \cdot 3 = 6$ ), resulting in $\frac{8x^3y^6}{z^3}$ .

ⓑ $\left(\frac{3ab^3}{c^2}\right)^4 = \frac{81a^4b^{12}}{c^8}$ : Use the Quotient to a Power Property to raise both the numerator and the denominator to the fourth power: $\frac{(3ab^3)^4}{(c^2)^4}$ . Apply the Product to a Power Property in the numerator: $\frac{3^4 a^4 (b^3)^4}{(c^2)^4}$ . Evaluate $3^4 = 81$ and use the Power Property to multiply the inner and outer exponents for both $b$ ( $3 \cdot 4 = 12$ ) and $c$ ( $2 \cdot 4 = 8$ ). The final simplified expression is $\frac{81a^4b^{12}}{c^8}$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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