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

Verified step by step guidance

1. Identify the rational exponent expression given: \(-3x^{\frac{1}{3}}\).
2. Recall that a rational exponent \(x^{\frac{m}{n}}\) can be rewritten as a radical expression \(\sqrt[n]{x^m}\).
3. In this case, the exponent is \(\frac{1}{3}\), which means the cube root of \(x\) raised to the first power: \(x^{\frac{1}{3}} = \sqrt[3]{x}\).
4. Apply the negative coefficient outside the radical: \(-3x^{\frac{1}{3}} = -3 \sqrt[3]{x}\).
5. Match this expression with the equivalent radical form \(-3 \sqrt[3]{x}\) in Column II.

Key Concepts

Rational Exponents

Rational exponents express roots and powers together, where the numerator is the power and the denominator is the root. For example, x^(1/3) means the cube root of x, and x^(m/n) means the nth root of x raised to the mth power.

Radical Expressions

Radical expressions use the root symbol (√) to represent roots of numbers or variables. The expression √[n]{x} denotes the nth root of x, which is equivalent to x raised to the power of 1/n in rational exponent form.

Negative Coefficients and Variables

When a negative coefficient is present, such as -3 in -3x^(1/3), it multiplies the entire expression. The negative sign affects the value but does not change the exponent or root operation on x.