Match the rational exponent expression in Column I with the equivalent radical expression in Column II. Assume that x is not 0. (b) ( -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: \(a^{m/n} = \sqrt[n]{a^m}\), where \(n\) is the root and \(m\) is the power.
2. Identify the base, exponent numerator, and denominator in the expression \(( -3x )^{-1/3}\). Here, the base is \(-3x\), the numerator of the exponent is \(-1\), and the denominator is \$3$.
3. Rewrite the expression using the radical form: \(( -3x )^{-1/3} = \left( \sqrt[3]{-3x} \right)^{-1}\), because the denominator \$3\( indicates a cube root and the numerator \)-1$ indicates the reciprocal.
4. Understand that raising to the power \(-1\) means taking the reciprocal, so \(\left( \sqrt[3]{-3x} \right)^{-1} = \frac{1}{\sqrt[3]{-3x}}\).
5. Therefore, the equivalent radical expression for \(( -3x )^{-1/3}\) is \(\frac{1}{\sqrt[3]{-3x}}\).

Key Concepts

Rational Exponents

Rational exponents express roots and powers simultaneously. An exponent in the form of a fraction a/b means raising the base to the power a and then taking the b-th root. For example, x^(m/n) equals the n-th root of x raised to the m-th power.

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. This rule helps simplify expressions with negative powers by rewriting them as fractions.

Converting Rational Exponents to Radical Expressions

To convert a rational exponent to a radical expression, rewrite x^(m/n) as the n-th root of x raised to the m-th power, √[n]{x^m}. For negative exponents, first rewrite as a reciprocal, then apply the root. This conversion clarifies the relationship between exponents and radicals.