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 \(a^{m/n}\) can be rewritten as a radical: \(a^{m/n} = \left( \sqrt[n]{a} \right)^m = \sqrt[n]{a^m}\).
2. Identify the base and the rational exponent in the expression \((-3x)^{-1/3}\). Here, the base is \(-3x\) and the exponent is \(-\frac{1}{3}\).
3. Rewrite the expression using the radical form: \((-3x)^{-1/3} = \frac{1}{(-3x)^{1/3}}\) because a negative exponent means the reciprocal.
4. Express \((-3x)^{1/3}\) as a cube root: \((-3x)^{1/3} = \sqrt[3]{-3x}\).
5. Combine the steps to write the original expression as \(\frac{1}{\sqrt[3]{-3x}}\), which is the equivalent radical expression.

Key Concepts

Rational Exponents

Rational exponents express roots and powers simultaneously, where the numerator indicates the power and the denominator indicates the root. For example, x^(m/n) means the nth root of x raised to the mth power, i.e., (√[n]{x})^m.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For instance, x^(-a) equals 1 divided by x^a, which flips the base to the denominator.

Converting Rational Exponents to Radical Expressions

To convert a rational exponent to a radical expression, rewrite x^(m/n) as the nth root of x raised to the mth power: x^(m/n) = (√[n]{x})^m. Applying this helps match expressions with radicals and rational exponents.