Problem 105 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[6]{x^{4}}$$

Step-by-Step Solution

Answer

The simplified form of the radical is \(x^{2/3}\).

Step 1: Understand the given radical

The given radical is \(\sqrt[6]{x^{4}}\), which means it's the 6th root of \(x^{4}\). The goal is to simplify this radical by reducing its index.

Step 2: Apply the power rule

In mathematics, the power rule is \(a^{m/n} =\sqrt[n]{a^{m}}\) where m and n are real numbers. Applying this rule to the given radical, we can rewrite \(\sqrt[6]{x^{4}}\) as \((x^{4})^{1/6}\).

Step 3: Simplify the exponent

For any exponent, \(a^{m/n} = a^{m \div n}\). When you multiply the exponents in \((x^{4})^{1/6}\) (i.e., 4 times 1/6), you get \(x^{4/6}\), which simplifies further to \(x^{2/3}\).