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

Verified step by step guidance

1. Recall that a rational exponent of the form \(a^{m/n}\) can be rewritten as a radical: \(a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\).
2. Identify the base and the exponent in the expression \((3x)^{-1/3}\). Here, the base is \((3x)\) and the exponent is \(-\frac{1}{3}\).
3. Apply the negative exponent rule: \(a^{-m} = \frac{1}{a^m}\). So, \((3x)^{-1/3} = \frac{1}{(3x)^{1/3}}\).
4. Rewrite \((3x)^{1/3}\) as the cube root of \$3x$: \((3x)^{1/3} = \sqrt[3]{3x}\).
5. Combine the steps to express \((3x)^{-1/3}\) as \(\frac{1}{\sqrt[3]{3x}}\), which is the equivalent radical expression.

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, (√[n]{x})^m. Understanding this allows conversion between exponential and radical forms.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-a) = 1/x^a. This concept is essential for rewriting expressions with negative rational exponents into radical form.

Properties of Radicals

Radical expressions involve roots, such as square roots or cube roots. Knowing how to express radicals as fractional exponents and manipulate them helps in matching and simplifying expressions like (3x)^(-1/3) into radical form.