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 \(a^{m/n}\) can be rewritten as a radical: \(a^{m/n} = \left( \sqrt[n]{a} \right)^m = \sqrt[n]{a^m}\).

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

Rewrite the expression using the radical form: \((-3x)^{-1/3} = \frac{1}{(-3x)^{1/3}}\) because a negative exponent means the reciprocal.

Express \((-3x)^{1/3}\) as a cube root: \((-3x)^{1/3} = \sqrt[3]{-3x}\).

Combine the steps to write the original expression as \(\frac{1}{\sqrt[3]{-3x}}\), which is the equivalent radical expression.

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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.

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