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Chapter 0: Problem 141

Factor completely. $$(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}}$$

Short Answer

Expert verified
The completely factored form of the expression is \(\frac{1}{\sqrt{x-5}}\left(\frac{1}{\sqrt{(x+5)}}-\sqrt{x+5}\right)\).

Step by step solution

01

Rewrite the expression

Firstly, rewrite the expression in terms of roots instead of fractional powers: \[\frac{1}{\sqrt{x-5}}\frac{1}{\sqrt{x+5}}-\sqrt{x+5}\frac{1}{(x-5)^{\frac{3}{2}}}\]
02

Change format and Identify common factors

Rearrange the terms so that common factors become clear and change the format: \[\frac{1}{\sqrt{(x-5)(x+5)}}-\frac{\sqrt{x+5}}{(x-5)\sqrt{x-5}}\] You can observe the common factor of \(\frac{1}{\sqrt{x-5}}\) between the two terms.
03

Factoring

Factor out the common factor of \(\frac{1}{\sqrt{x-5}}\) from both terms. After pulling out the common factor, factor the expression completely: \[\frac{1}{\sqrt{x-5}}\left(\frac{1}{\sqrt{(x+5)}}-\sqrt{x+5}\right)\]

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