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 July 14th, 2012, 02:35 PM #1 Senior Member   Joined: Jan 2010 Posts: 205 Thanks: 0 How to simplify algebraic expression $\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}(\frac{x - 3}{x - 2})^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}}$ $\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}(\frac{x - 3}{x - 2})^{-\frac{1}{2}} \cdot \frac{1}{(x - 2)^{2}} \\\\\\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2}\frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\\\\\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{\frac{1}{2}}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\\\\\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(x - 2)^{2}} \\\\\\ \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{(\frac{x - 3}{x - 2})^{\frac{1}{2}}} \cdot \frac{1}{2(x - 2)^{2}} \\\\\\ \frac{1}{(\frac{x - 3}{x - 2})} \cdot \frac{1}{2(x - 2)^{2}} \\\\\\ \frac{1}{(x - 3)} \cdot \frac{1}{2(x - 2)} \\\\\\ \frac{1}{2(x - 3)(x - 2)} \\\\\\$ Which step have I done incorrectly?
 July 14th, 2012, 03:57 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,975 Thanks: 1157 Math Focus: Elementary mathematics and beyond Re: How to simplify algebraic expression None, that is correct.
 July 14th, 2012, 07:54 PM #3 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 Re: How to simplify algebraic expression Daigo, tired me out just looking at all that "writing"; on a timed test, you're in trouble! I suggest you always look for short cuts; like: a = x-2, b = x - 3 ; then expression shortens to: 1 / (b/a)^(1/2) * (b/a)^(-1/2) / 2 * 1 / a^2 = (b/a)^(-1/2) / [2a^2 * (b/a)^(1/2)] = 1 / [2a^2 * (b/a)^(1/2) * (b/a)^(1/2)] = 1 / [2a^2 * (b/a)] = 1 / (2ab) = 1 / [2(x-1)(x-3)] Got that? Btw, good job in getting correct answer...I'm impressed
 July 14th, 2012, 08:29 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs Re: How to simplify algebraic expression I would just write (doing a few simple operations mentally): $\frac{1}{(\frac{x-3}{x-2})^{\small{\frac{1}{2}}}}\cdot\frac{1}{2}$$\frac{ x-3}{x-2}$$^{\small{-\frac{1}{2}}}\cdot\frac{1}{(x-2)^{\small{2}}}=\frac{1}{2(x-3)(x-2)}$ We see that the factor with the negative exponent will combine with the denominator of the first factor to give the rational expression there an exponent of 1, and then its denominator will cancel with the denominator of the 4th factor, leaving the end result.
 July 15th, 2012, 01:55 AM #5 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: How to simplify algebraic expression $\frac{1}{$$\frac{x-3}{x-2}$$^{\frac{1}{2}}} \cdot \frac{1}{2} \cdot $$\frac{x-3}{x-2}$$^{-\frac{1}{2}} \cdot \frac{1}{(x-2)^{2}}$ $= \frac{1}{2} \cdot \frac{1}{$$\frac{x-3}{x-2}$$^{\frac{1}{2}} \cdot $$\frac{x-3}{x-2}$$^{\frac{1}{2}}} \cdot \frac{1}{(x-2)^{2}}$ $= \frac{1}{2} \cdot \frac{1}{$$\frac{x-3}{x-2}$$} \cdot \frac{1}{(x-2)^{2}}$ $= \frac{1}{2} \cdot \frac{x-2}{x-3} \cdot \frac{1}{(x-2)^{2}}$ $= \frac{1}{2} \cdot \frac{\cancel{x-2}}{x-3} \cdot \frac{1}{x-2 \cdot \cancel{x-2}} = \frac{1}{2(x-3)(x-2)}$ $\text{QED}$

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