Problem 106 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[9]{x^{6}}$$

Step-by-Step Solution

Answer

The simplification of \(\sqrt[9]{x^{6}}\) by reducing the index of the radical is \(\sqrt[3]{x^{2}}\).

Step 1: Identify the Index and Exponent

In this case, the index of the radical is 9 and the exponent of \(x\) is 6.

Step 2: Find Common Divisor

Find an integer that divides evenly into both the exponent and the index. The greatest common divisor (GCD) of 6 and 9 is 3.

Step 3: Reduce the index of the radical

Divide both the exponent and the index by the GCD. Divide 6 by 3 to get 2, and divide 9 by 3 to get 3. So, the simplified radical is \(\sqrt[3]{x^{2}}\).