1Cademy - Simplifying $$(x^4)^{\frac{1}{2}}$$, $$(y^6)^{\frac{1}{3}}$$, and $$(z^9)^{\frac{2}{3}}$$ Using the Power Property with Rational Exponents

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Power Property for Exponents

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Simplifying $(x^4)^{\frac{1}{2}}$ , $(y^6)^{\frac{1}{3}}$ , and $(z^9)^{\frac{2}{3}}$ Using the Power Property with Rational Exponents

Apply the Power Property for Exponents to simplify three expressions in which a variable raised to an integer power is then raised to a rational (fractional) exponent. The rule $(a^m)^n = a^{m \cdot n}$ works the same way when $n$ is a fraction — multiply the exponents and simplify.

ⓐ $(x^4)^{\frac{1}{2}} = x^2$ : The inner exponent is $4$ and the outer exponent is $\frac{1}{2}$ . Multiply the exponents using the Power Property: $(x^4)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}}$ . Simplify the product: $4 \cdot \frac{1}{2} = 2$ , so the result is $x^2$ .

ⓑ $(y^6)^{\frac{1}{3}} = y^2$ : The inner exponent is $6$ and the outer exponent is $\frac{1}{3}$ . Multiply: $(y^6)^{\frac{1}{3}} = y^{6 \cdot \frac{1}{3}}$ . Simplify: $6 \cdot \frac{1}{3} = 2$ , giving $y^2$ .

ⓒ $(z^9)^{\frac{2}{3}} = z^6$ : The inner exponent is $9$ and the outer exponent is $\frac{2}{3}$ . Multiply: $(z^9)^{\frac{2}{3}} = z^{9 \cdot \frac{2}{3}}$ . Simplify: $9 \cdot \frac{2}{3} = \frac{18}{3} = 6$ , giving $z^6$ .

In each case the procedure is identical to raising a power to an integer power — keep the base and multiply the exponents. With a fractional outer exponent, the multiplication step involves multiplying an integer by a fraction. Parts ⓐ and ⓑ use exponents of the form $\frac{1}{n}$ , while part ⓒ uses a general rational exponent $\frac{m}{n}$ , showing that the Power Property applies uniformly to all rational exponents.

Updated 2026-05-01

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