Problem 141 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite the expression

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

Step 2: 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.

Step 3: 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: \[\frac{1}{\sqrt{x-5}}\left(\frac{1}{\sqrt{(x+5)}}-\sqrt{x+5}\right)\]