1Cademy - Writing $$x^{\frac{1}{2}}$$, $$y^{\frac{1}{3}}$$, and $$z^{\frac{1}{4}}$$ as Radical Expressions

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

Example

Writing $x^{\frac{1}{2}}$ , $y^{\frac{1}{3}}$ , and $z^{\frac{1}{4}}$ as Radical Expressions

Convert three expressions with rational exponents of the form $\frac{1}{n}$ into radical notation by using the rule $a^{\frac{1}{n}} = \sqrt[n]{a}$ .

ⓐ $x^{\frac{1}{2}} = \sqrt{x}$ : The denominator of the exponent is $2$ , so the index of the radical is $2$ . By convention, the index is not written when it equals $2$ — the expression is simply $\sqrt{x}$ .

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

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

In each case, the conversion follows the same pattern: the base of the exponent becomes the radicand, and the denominator of the fractional exponent becomes the index of the radical. When the index is $2$ , the radical is a square root and the index is omitted by convention.

Updated 2026-05-01

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