1Cademy - Simplifying $$\left(\frac{4p^{-3}}{q^2}\right)^2$$ and $$\left(\frac{3x^{-2}}{y^3}\right)^3$$ Using Exponent Properties

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Simplifying $(5x^{-3})^2$ and $(8a^{-4})^2$ Using Exponent Properties

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Simplifying $\left(\frac{4p^{-3}}{q^2}\right)^2$ and $\left(\frac{3x^{-2}}{y^3}\right)^3$ Using Exponent Properties

Combine the Quotient to a Power Property, the Product to a Power Property, the Power Property, and the definition of a negative exponent to simplify complex rational expressions.

ⓐ $\left(\frac{4p^{-3}}{q^2}\right)^2 = \frac{16}{p^6q^4}$ : Start by applying the Quotient to a Power Property to distribute the outer exponent $2$ : $\frac{(4p^{-3})^2}{(q^2)^2}$ . Use the Product to a Power Property in the numerator: $\frac{4^2 (p^{-3})^2}{(q^2)^2}$ . Apply the Power Property to multiply exponents, yielding $\frac{16p^{-6}}{q^4}$ . Finally, use the definition of a negative exponent ( $a^{-n} = \frac{1}{a^n}$ ) to rewrite $p^{-6}$ in the denominator with a positive exponent, resulting in $\frac{16}{p^6q^4}$ .

ⓑ $\left(\frac{3x^{-2}}{y^3}\right)^3 = \frac{27}{x^6y^9}$ : Apply the Quotient to a Power Property: $\frac{(3x^{-2})^3}{(y^3)^3}$ . Distribute the outer exponent in the numerator using the Product to a Power Property: $\frac{3^3 (x^{-2})^3}{(y^3)^3}$ . Multiply exponents using the Power Property and evaluate the coefficient: $\frac{27x^{-6}}{y^9}$ . Move the factor with the negative exponent to the denominator to complete the simplification: $\frac{27}{x^6y^9}$ .

Updated 2026-05-26

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OpenStax Intermediate Algebra

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