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. 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 to the term \(x^{-\frac{1}{3}}\): rewrite the exponent as a radical with a cube root, so \(x^{-\frac{1}{3}} = \left(\sqrt[3]{x}\right)^{-1}\).
4. Understand that raising to the power of \(-1\) means taking the reciprocal, so \(\left(\sqrt[3]{x}\right)^{-1} = \frac{1}{\sqrt[3]{x}}\).
5. Combine the coefficient 3 with the radical expression to write the equivalent radical form: \(3 \times \frac{1}{\sqrt[3]{x}} = \frac{3}{\sqrt[3]{x}}\).

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. 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 instance, x^(-a) equals 1 divided by x^a. This concept is essential for rewriting expressions with negative rational exponents into radical form.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, represented with the radical symbol (√). Converting rational exponents to radicals requires recognizing that x^(1/n) equals the nth root of x, which helps in matching expressions between exponential and radical forms.