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How do I simplify the following types of question:

$ \sqrt{x^2+5} \times \sqrt{x^2+20}$

Do I need to get both answer out of their roots first or not? This is how I would do it:

$ \sqrt{x^2+5} \times \sqrt{x^2+20} = x^2 + x\sqrt{20} + x\sqrt{5} + \sqrt{100}$ but I'm very certain this isn't the correct way.

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Just a brief note about vocabulary: $\sqrt{x^2 + 5} \cdot \sqrt{x^2 + 20}$ can not be solved, since there is no equation. Presumably, you mean to be ask how to simplify the expression. –  JavaMan Feb 14 '13 at 18:29
    
@JavaMan Thanks –  Birdistheword Feb 14 '13 at 18:34

1 Answer 1

up vote 1 down vote accepted

For positive real $a,b,c,d$

$\sqrt{a+b}\cdot\sqrt{c+d}=\sqrt{(a+b)(c+d)}=\sqrt{ac+ad+bc+bd}$ which is $\ne \sqrt{ac}+\sqrt{ad}+\sqrt{bc}+\sqrt{bd}$

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That sounds more logical than what I thought of –  Birdistheword Feb 14 '13 at 18:38
    
@Birdistheword, observe that the last inequality always holds for positive real numbers. –  lab bhattacharjee Feb 14 '13 at 18:40

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