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Chapter 0: Problem 100

Factor and simplify each algebraic expression. $$\left(x^{2}+3\right)^{-\frac{2}{3}}+\left(x^{2}+3\right)^{-\frac{5}{3}}$$

Short Answer

Expert verified
The factored and simplified form of the expression is \( \left(x^{2}+3\right)^{-\frac{2}{3}} \times (1+ \left(x^{2}+3\right)^{-1} ) \).

Step by step solution

01

Identify common factors

The two terms \( \left(x^{2}+3\right)^{-\frac{2}{3}} \) and \( \left(x^{2}+3\right)^{-\frac{5}{3}} \) both have \( \left(x^{2}+3\right)^{-\frac{2}{3}} \) as a factor.
02

Rewrite the expression

Factor the expression by pulling out the common factor \( \left(x^{2}+3\right)^{-\frac{2}{3}} \), to get it in the format \[ 1+\left(x^{2}+3\right)^{-1} \]. This is done by subtracting the powers of the common factor in each term.
03

Simplify

Rewrite simplified expression as \( \left(x^{2}+3\right)^{-\frac{2}{3}} \times (1+ \left(x^{2}+3\right)^{-1} ) \).

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Most popular questions from this chapter

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Explain how to multiply rational expressions.

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Explain how to simplify a rational expression.

Will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.
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