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I'm having a problems simplifying this apparently simple radical: $\sqrt{\frac{1}{a}}$

The book I'm working through gives the answer as: $\frac{1}{a}\sqrt{a}$

Could someone break down the steps used to get there? I've managed to answer all the other questions in this chapter right, but my brain refuses to lock onto this one and I'm feeling really dense.

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I assume that $a \gt 0$. Expand the fraction $\frac{1}{a}$ by $a$ to get $\frac{a}{a^2}$. Then use $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ and $\sqrt{a^2} = a$. –  t.b. Apr 29 '11 at 3:11
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Help me here... how is $\frac1{\sqrt{a}}$ less simple than $\frac{\sqrt{a}}{a}$? (I'm kidding; I just don't understand why they don't say "rationalize" instead...) –  J. M. Apr 29 '11 at 3:13

3 Answers 3

up vote 5 down vote accepted

$\sqrt{1/a} = 1/\sqrt{a} = (1/\sqrt{a})(\sqrt{a}/\sqrt{a}) = \sqrt{a}/a$ for $a > 0$. I have these brain failures sometimes too, no need to feel too dense.

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Thanks guys. I managed to get $\sqrt{\frac{2}{7}}$ to $\frac{1}{7}\sqrt{14}$ just fine, but they tossed that letter at me and my brain froze apparently. –  DaveG Apr 29 '11 at 3:53

Multiply the numerator and the denominator by $\sqrt{a}$. You can do this because dividing and multiplying by the same nonzero number does not change the result.

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What do you get if you square both expressions and then simplify?

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