How to simplify algebraic expression

**daigo** · July 14th 2012, 01:52 PM

$\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}(\frac{x - 3}{x - 2})^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}}$

$\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}(\frac{x - 3}{x - 2})^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}} \\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{\frac{1}{2}}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2(x - 2)^{2}} \\ \frac{1}{(\frac{x - 3}{x - 2})} \cdot \frac{1}{2(x - 2)^{2}} \\ \frac{1}{(x - 3)} \cdot \frac{1}{2(x - 2)} \\ \frac{1}{2(x - 3)(x - 2)}$

Which step have I done incorrectly?

**Soroban** · July 14th 2012, 02:37 PM

Hello, daigo!

Your answer is correct.
You made a few unnecessary steps, though.

$\frac{1}{\left(\dfrac{x - 3}{x - 2}\right)^{\frac{1}{2}}} \cdot \frac{1}{2}\left(\frac{x - 3}{x - 2}\right)^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}}$

$\text{We have: }\;\frac{1}{2}\cdot\underbrace{\frac{1}{\left( \dfrac{x-3}{x-2}\right)^{\frac{1}{2}}}\cdot \frac{1}{\left(\dfrac{x-3}{x-2}\right)^{\frac{1}{2}}}}_{\downarrow} \cdot\frac{1}{(x-2)^2}$

. . . . . . . . . . . . $=\;\frac{1}{2}\cdot\frac{1}{\frac{x-3}{x-2}} \cdot \frac{1}{(x-2)^2}$

. . . . . . . . . . . . $=\;\frac{1}{2(x-3)(x-2)}$

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