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

Verified step by step guidance

1. Recall that a rational exponent of the form \(a^{\frac{m}{n}}\) can be rewritten as a radical expression: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\).
2. Identify the base and the exponent in the expression \((3x)^{\frac{1}{3}}\). Here, the base is \((3x)\) and the exponent is \(\frac{1}{3}\).
3. Apply the rule for rational exponents: \((3x)^{\frac{1}{3}} = \sqrt[3]{(3x)^1}\), which simplifies to the cube root of \$3x$.
4. Write the equivalent radical expression as \(\sqrt[3]{3x}\), which means the cube root of the product \$3x$.
5. Confirm that this radical expression matches the one given in Column II to complete the matching process.

Key Concepts

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.