December 28th, 2014, 06:32 AM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  Simplify
How to simplify this expression? $\displaystyle \frac{1}{x}\cdot \sqrt{4+3x}+\frac{3}{2\sqrt{3x+4}}\cdot\ln x$ I know the answer is $\displaystyle \frac{3x(\ln(x)+2)+8}{2x\sqrt{3x+4}}$. But how? Last edited by skipjack; December 28th, 2014 at 07:48 PM. 
December 28th, 2014, 07:15 AM  #2 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177  The common denominator is $\displaystyle \ 2x\sqrt{3x+4}$ The first fraction is amplified by $\displaystyle \ 2\sqrt{3x+4}$ and the second fraction is amplified by $\displaystyle \ x$ Thus, we get $\displaystyle \color{blue}{\dfrac{2\sqrt{3x+4}\cdot\sqrt{3x+4}+3 x \ln x}{2x \sqrt{3x+4}}=\dfrac{2(3x+4)+3x \ln x}{2x \sqrt{3x+4}}=\dfrac{2\cdot3x+8+3x \ln x}{2x \sqrt{3x+4}} \\\;\\ =\dfrac{3x(2+ \ln x) + 8}{2x \sqrt{3x+4}}}$ Last edited by skipjack; December 28th, 2014 at 07:51 PM. 

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