Problem 73 - College Algebra

Question

Simplify the radical expressions if possible. $$\frac{\sqrt[5]{64 x^{6}}}{\sqrt[5]{2 x}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(2x\).

Step 1: Identify similar radical components

Notice that both the numerator and the denominator have \(\sqrt[5]{2x}\) as a component. This will be used to simplify the expression.

Step 2: Divide similar components

Divide \(\sqrt[5]{64x^{6}}\) by \(\sqrt[5]{2x}\). The denominator of \(\sqrt[5]{2x}\) can be seen as \(\sqrt[5]{64x^{6}}\) powered to the 1/6. Recalling that when you divide expressions with the same base, you subtract the exponents, the simplified form is \(\sqrt[5]{64x^{6}} / \sqrt[5]{2x} = \sqrt[5]{(64x^{6})^{1-1/6}}\). Simplifying this gives \(\sqrt[5]{32x^{5}}\).

Step 3: Final simplification

Lastly, recognizing that the fifth root of 32x^5 yields simple numbers, the expression can be further simplified as \(\sqrt[5]{32x^{5}} = 2x\) to finalize the solution.