1Cademy - Writing $$(\sqrt[3]{2x})^4$$ and $$\sqrt{\left(\frac{3a}{4b}\right)^3}$$ with Rational Exponents

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

Example

Writing $(\sqrt[3]{2x})^4$ and $\sqrt{\left(\frac{3a}{4b}\right)^3}$ with Rational Exponents

Convert radical expressions with compound radicands (products or fractions) and outer exponents into rational exponent form using the definitions $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ and $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ .

ⓐ $(\sqrt[3]{2x})^4 = (2x)^{\frac{4}{3}}$ : The entire radicand $2x$ is the base. The index is $3$ (which becomes the denominator) and the outer power is $4$ (which becomes the numerator). Because the base is a product, it must be enclosed in parentheses to show the exponent applies to both factors: $(2x)^{\frac{4}{3}}$ .

ⓑ $\sqrt{\left(\frac{3a}{4b}\right)^3} = \left(\frac{3a}{4b}\right)^{\frac{3}{2}}$ : The base is the fraction $\frac{3a}{4b}$ . The index is $2$ (an unwritten square root) and the power is $3$ . The entire fraction becomes the base raised to the $\frac{3}{2}$ power: $\left(\frac{3a}{4b}\right)^{\frac{3}{2}}$ .

When the base consists of multiple factors or a fraction, parentheses are required to indicate that the rational exponent applies to the entire expression.

Updated 2026-05-01

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OpenStax Intermediate Algebra

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