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Express $\sqrt{x} \div\sqrt{y}$ as a common fraction, given: $\frac{ {\left( \frac{1}{2} \right)}^2 + {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 + {\left( \frac{1}{5} \right)}^2} = \frac{13x}{41y} $

 Feb 18, 2022
 #1
avatar+61 
+3

The first thing that I did was simplify ((1/2)^2+(1/3)^2)/((1/4)^2+(1/5)^2), which is (13/36)/(41/400). This can be simplified so that it looks like this: (13/41)(100/9). x/y=100/9, so \(\sqrt{x}/\sqrt{y}=10/3\)

 Feb 18, 2022
 #2
avatar+61 
+3

I'm not really good at expressing things with the LaTeX thing, so I apologize if my answer is unclear, I can try it if you want.

Ooflord  Feb 18, 2022
 #3
avatar+61 
+3

First, I simplify \(\frac{ {\left( \frac{1}{2} \right)}^2 + {\left( \frac{1}{3} \right)}^2 }{ {\left( \frac{1}{4} \right)}^2 + {\left( \frac{1}{5} \right)}^2} = \frac{13x}{41y} \) so that it looks like \(\frac{ {\left(\frac{1}{25} \right)} + {\left( \frac{1}{9} \right)} }{ {\left( \frac{1}{16} \right)} + {\left( \frac{1}{25} \right)}} = \frac{13x}{41y} \), and then I added the fractions on the numerator and the fractions on the denominator so I got \((\frac{13}{36})/(\frac{41}{400})\), which can then be simplified to \(13*100/41*9\).

remember that this is equal to \(13x/41y\), so next, I get rid of 13/41 on both sides, giving me \(x/y=100/9\). I want \(\sqrt{x}/\sqrt{y}\), so I square root both sides of \(x/y=100/9\), which gets me\(\sqrt{x}/\sqrt{y}=10/3\).

Ooflord  Feb 18, 2022

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