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Chapter 0: Problem 105

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

Short Answer

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

Step by step solution

01

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.
02

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 \((x^{4})^{1/6}\).
03

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 \(x^{2/3}\).

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Most popular questions from this chapter

Perform the indicated operations. $$ \left(x^{n}+2\right)\left(x^{n}-2\right)-\left(x^{n}-3\right)^{2} $$

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$ \frac{1}{2}+\frac{2}{3} $$

Perform the indicated operations. $$ [(3 x+y)+1]^{2} $$

If \(b^{A}=M N, b^{C}=M,\) and \(b^{D}=N,\) what is the relationship among \(A, C,\) and \(D ?\)

Simplify using properties of exponents. $$\left(3 x^{\frac{2}{3}}\right)\left(4 x^{\frac{3}{4}}\right)$$
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