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As homework I was asked to simply the following radical expression.

$\sqrt{3ax^2+18ax+27a}$

Is following the farthest I can simplify the above expression ?

$\sqrt{3a(x^2+6x+9)}$

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$$x^2 + 6x + 9 = (x+3)^2{}{}{}{}{}{}$$

That gives you $$\sqrt{3a(x^2+6x+9)} = \sqrt{3a(x+3)^2} = |x+3|\sqrt{3a}$$

amWhy
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  • Thank you. Just a small question. why has you written $|x+3|$ instead of $(x+3)$ ? –  Feb 18 '14 at 13:14
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    The expression in the radical will necessarily evaluate to a non-negative number, assuming $a$ is nonzero, because $(x + 3)^2 \geq 0$. And the square root function returns the positive root. So whatever $x$ is "plugged in", the expression will evaluate to a non-negative. We need to make sure this is also the case with the final expression. For example, suppose $a = 1, x = -9$. Compare the original expression, evaluated at $a = 1, x = -9$, with the simplified expression, if we left out the absolute value sign around $x + 3$. – amWhy Feb 18 '14 at 13:19
  • It pretty much makes sense. Thanks ! –  Feb 18 '14 at 13:30
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    You're welcome. It might be easier to understand if we simply look at $\sqrt{x^2} = |x|$. Suppose $x = -2$. Then $\sqrt{x^2} = \sqrt{(-2)^2} = \sqrt 4 = 2 = |(-2)|$. – amWhy Feb 18 '14 at 13:33