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Disclosure: This is homework, but not part of the homework. This is just something that I do not understand.

$ x = \sqrt{\frac{5}{3}} $

$ x = \frac{\sqrt{15}}{3} $

Could anyone please explain this to me?

Thanks in advance.

  • 2
    If you only want to *verify* that the two are the same, (note that both are positive and) square them: $x^2$ in the former is $\displaystyle \frac{5}{3}$ and in the latter is $\displaystyle \frac{15}{9}$ and you probably know how to verify that the two are the same.2011-11-13

3 Answers 3

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$ x= \sqrt{\frac{5}{3}} = \frac{\sqrt{5}}{\sqrt{3}} =\frac{\sqrt{5} \times\sqrt{3} }{\sqrt{3}\times \sqrt{3}} = \frac{\sqrt{15}}{3}$

This is called rationalizing the denominator, you can practice more here.

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If you have the root $\sqrt{5/3}$, you can simply extend by three, yielding $\sqrt{15/9}$. Then you can proceed by the laws for roots and get $\sqrt{15\over9} = \frac{\sqrt{15}}{\sqrt9} = \frac{\sqrt{15}}3$

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    @Max ${5\over3}\to{15\over9}$2011-11-13
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Multiply top and bottom of the fraction by $\sqrt{3}$ and you get $\sqrt{3} \cdot \sqrt{5}=\sqrt{15}$ on top and $\sqrt{3} \cdot \sqrt{3}=3$ on the bottom. The trick is to multiply the fraction by the bottom square root, thus getting rid of the square root in the bottom of the fraction. Mathematicians don't like square roots on the bottom of fraction :)

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    Sorry if I came across as picky. I didn't intend to fault you when I wrote that comment; just clarifying my stand. =)2011-11-13