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 it as \(\left(\sqrt[3]{x}\right)^{-1}\), which is the reciprocal of the cube root of \(x\).
4. Since the exponent is negative, rewrite \(x^{-\frac{1}{3}}\) as \(\frac{1}{x^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{x}}\).
5. Combine the coefficient \(-3\) with the radical expression to get the equivalent radical form: \(-3 \cdot \frac{1}{\sqrt[3]{x}} = \frac{-3}{\sqrt[3]{x}}\).

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, or (√[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}. This helps in matching expressions involving radicals and rational exponents.