Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. ( 3x )^1/3

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Recall that a rational exponent of the form $a^{\frac{m}{n}}$ can be rewritten as a radical expression: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Identify the base and the exponent in the expression $(3x)^{\frac{1}{3}}$. Here, the base is $(3x)$ and the exponent is $\frac{1}{3}$.

Apply the rule for rational exponents: $(3x)^{\frac{1}{3}} = \sqrt[3]{(3x)^1}$, which simplifies to the cube root of \$3x$.

Write the equivalent radical expression as $\sqrt[3]{3x}$, which means the cube root of the product \$3x$.

Confirm that this radical expression matches the one given in Column II to complete the matching process.

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Rational Exponents

Rational exponents represent roots and powers combined. An expression like x^(m/n) means the nth root of x raised to the mth power, or equivalently, (x^(1/n))^m. Understanding this allows conversion between exponent and radical forms.

Radical Expressions

Radical expressions use root symbols to denote roots, such as square roots or cube roots. The nth root of a number x is written as √[n]{x}, which corresponds to x raised to the 1/n power. Recognizing this helps match radicals to rational exponents.

Properties of Exponents

Properties of exponents include rules like (ab)^m = a^m * b^m and (x^m)^n = x^(mn). These rules allow simplification and rewriting of expressions involving powers and roots, essential for matching expressions like (3x)^(1/3) to their radical equivalents.

Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. ( 3x )1/3