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Chapter 0: Problem 73

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

Short Answer

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

Step by step solution

01

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.
02

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 \(\sqrt[5]{32x^{5}}\).
03

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.

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