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

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Rational Exponent $a^{\frac{1}{n}}$

Example

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

Convert three radical expressions into rational exponent form by applying the equivalence $\sqrt[n]{a} = a^{\frac{1}{n}}$ in the radical-to-exponent direction.

ⓐ $\sqrt{x} = x^{\frac{1}{2}}$ : When a square root has no index written, the index is $2$ by convention. The index becomes the denominator of the rational exponent, so $\sqrt{x} = x^{\frac{1}{2}}$ .

ⓑ $\sqrt[3]{y} = y^{\frac{1}{3}}$ : The index is $3$ , so the denominator of the exponent is $3$ . The expression becomes $y^{\frac{1}{3}}$ .

ⓒ $\sqrt[4]{z} = z^{\frac{1}{4}}$ : The index is $4$ , so the denominator of the exponent is $4$ . The expression becomes $z^{\frac{1}{4}}$ .

In each case, the conversion follows the same pattern: the radicand becomes the base, and the index of the radical becomes the denominator of the fractional exponent $\frac{1}{n}$ . This is the reverse of writing a rational exponent as a radical — both directions use the same rule $a^{\frac{1}{n}} = \sqrt[n]{a}$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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