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

Verified step by step guidance

1. Identify the given expression: \$3x^{-\frac{1}{3}}$.
2. Recall that a rational exponent \(x^{\frac{m}{n}}\) can be rewritten as a radical: \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\).
3. Apply this rule to the term \(x^{-\frac{1}{3}}\): rewrite it as \(\sqrt[3]{x^{-1}}\).
4. Remember that a negative exponent means the reciprocal, so \(\sqrt[3]{x^{-1}} = \frac{1}{\sqrt[3]{x}}\).
5. Combine the coefficient 3 with the rewritten radical expression to get the equivalent radical form: \(\frac{3}{\sqrt[3]{x}}\).

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 means x^(-1/3) is 1 over the cube root of x.

Converting Between Radical and Exponential Forms

Expressions with rational exponents can be rewritten as radicals and vice versa. For example, x^(1/3) is equivalent to the cube root of x (∛x), and x^(-1/3) is the reciprocal of the cube root of x, facilitating easier manipulation and understanding.