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Chapter 0: Problem 95

Factor and simplify each algebraic expression. $$4 x^{-\frac{2}{3}}+8 x^{\frac{1}{3}}$$

Short Answer

Expert verified
The simplified form of the algebraic expression is \(4\sqrt[3]{x}\left(\frac{1}{x} + 2\right)\)

Step by step solution

01

Identify Common Factors

In the given expression \(4 x^{-\frac{2}{3}}+8 x^{\frac{1}{3}}\), the common factor between the two terms present is \(4x^{\frac{1}{3}}\). Both terms are divisible by this factor.
02

Factor out the Common Factor

Now, divide both terms by the identified common factor, which gives: \(4x^{\frac{1}{3}}\left(x^{-1} + 2\right)\).
03

Simplify the Expression

The expression inside the parenthesis can be simplified further, changing the exponent -1 to a fraction in the denominator gives \(4x^{\frac{1}{3}}\left(\frac{1}{x} + 2\right)\). And for \(x\), using the property \(x^{\frac{1}{3}} = \sqrt[3]{x}\) gives the final simplification: \(4\sqrt[3]{x}\left(\frac{1}{x} + 2\right)\).

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