1Cademy - Simplifying $$\frac{(x^3)^4(x^{-2})^5}{(x^6)^5}$$ Using Several Exponent Properties

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Power Property for Exponents

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Simplifying $\frac{(x^3)^4(x^{-2})^5}{(x^6)^5}$ Using Several Exponent Properties

To simplify the rational expression $\frac{(x^3)^4(x^{-2})^5}{(x^6)^5}$ , apply the Power Property, Product Property, and Quotient Property in sequence. First, use the Power Property $(a^m)^n = a^{m \cdot n}$ on the numerator and the denominator: $(x^3)^4 = x^{12}$ , $(x^{-2})^5 = x^{-10}$ , and $(x^6)^5 = x^{30}$ . The expression becomes $\frac{(x^{12})(x^{-10})}{x^{30}}$ . Next, add the exponents in the numerator: $x^{12} \cdot x^{-10} = x^2$ . The expression simplifies to $\frac{x^2}{x^{30}}$ . Finally, use the Quotient Property $\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ to subtract the exponents: $\frac{1}{x^{30-2}} = \frac{1}{x^{28}}$ .

Updated 2026-05-26

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OpenStax Intermediate Algebra

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