1Cademy - Writing $$\sqrt{y^3}$$, $$\sqrt[3]{x^2}$$, and $$\sqrt[4]{z^3}$$ with Rational Exponents

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Rational Exponent $a^{\frac{m}{n}}$

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Writing $\sqrt{y^3}$ , $\sqrt[3]{x^2}$ , and $\sqrt[4]{z^3}$ with Rational Exponents

Convert three radical expressions — each containing a variable raised to a power inside the radical — into rational exponent notation using the rule $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . ⓐ $\sqrt{y^3} = y^{\frac{3}{2}}$ : The radicand is $y^3$ and the index is $2$ (square root). The exponent on $y$ inside the radical becomes the numerator of the rational exponent, and the index becomes the denominator: $\sqrt{y^3} = y^{\frac{3}{2}}$ . ⓑ $\sqrt[3]{x^2} = x^{\frac{2}{3}}$ : The radicand is $x^2$ and the index is $3$ (cube root). The exponent $2$ on $x$ becomes the numerator, and the index $3$ becomes the denominator: $\sqrt[3]{x^2} = x^{\frac{2}{3}}$ . ⓒ $\sqrt[4]{z^3} = z^{\frac{3}{4}}$ : The radicand is $z^3$ and the index is $4$ (fourth root). The exponent $3$ on $z$ becomes the numerator, and the index $4$ becomes the denominator: $\sqrt[4]{z^3} = z^{\frac{3}{4}}$ . In every case, the conversion follows the same pattern: the exponent on the variable inside the radical becomes the numerator of the fractional exponent, and the index of the radical becomes the denominator.

Updated 2026-05-01

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