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^{1/3}\(. Here, the exponent \)1/3$ is a rational exponent.
2. Recall the rule that a rational exponent \(x^{m/n}\) can be rewritten as a radical: \(x^{m/n} = \sqrt[n]{x^m}\).
3. Apply this rule to \(x^{1/3}\), which means \(m = 1\) and \(n = 3\), so \(x^{1/3} = \sqrt[3]{x^1} = \sqrt[3]{x}\).
4. Rewrite the entire expression \$3x^{1/3}$ as \(3 \times \sqrt[3]{x}\), since the coefficient 3 remains as a multiplier.
5. Thus, the equivalent radical expression for \$3x^{1/3}$ is \(3 \sqrt[3]{x}\).

Key Concepts

Rational Exponents

Rational exponents express roots and powers simultaneously. An exponent in the form of a fraction a/b means the b-th root of the base raised to the a-th power, such as x^(1/3) representing the cube root of x.

Radical Expressions

Radical expressions use root symbols to denote roots of numbers or variables. For example, the cube root of x is written as ∛x, which is equivalent to x raised to the 1/3 power.

Equivalence Between Rational Exponents and Radicals

Rational exponents and radicals are two ways to represent the same mathematical operation. Specifically, x^(m/n) is equivalent to the n-th root of x raised to the m-th power, allowing conversion between exponential and radical forms.