Question 249026
Here's one way to do it



{{{sqrt(5)/sqrt(180)}}} Start with the given expression.



{{{sqrt(5/180)}}}  Combine the roots using the identity {{{sqrt(x)/sqrt(y)=sqrt(x/y)}}}



{{{sqrt(1/36)}}} Reduce the fraction.



{{{sqrt(1)/sqrt(36)}}} Break up the fraction using the identity {{{sqrt(x/y)=sqrt(x)/sqrt(y)}}} (ie it's the reverse of the other identity)



{{{1/sqrt(36)}}} Take the square root of 1 to get 1.



{{{1/6}}} Take the square root of 36 to get 6.



So {{{sqrt(5)/sqrt(180)=1/6}}}




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Or...


Here's another way to do it. This method is more common as it works better in general.



{{{sqrt(5)/sqrt(180)}}} Start with the given expression.



The goal is to get the denominator equal to a rational number. To do this, we multiply the denominator by {{{sqrt(180)}}}. We must do the same for the numerator as well.


{{{(sqrt(5)*sqrt(180))/(sqrt(180)*sqrt(180))}}} Multiply both the numerator and denominator by {{{sqrt(180)}}}



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Scratch Work</font>



Take note that {{{sqrt(180)*sqrt(180)=sqrt(180*180)}}} since {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}} (it's like those previous identities). Since {{{sqrt(180)*sqrt(180)=sqrt(180*180)}}}, we can say that {{{sqrt(180)*sqrt(180)=sqrt(180*180)=sqrt(180^2)=180}}}. 


For the last step of that equation, the square root 'undoes' the square. In other words, {{{sqrt(x^2)=x}}} where {{{x>=0}}}



In the end, {{{sqrt(180)*sqrt(180)=180}}}


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Now back to the problem at hand....



{{{(sqrt(5)*sqrt(180))/180}}} Multiply {{{sqrt(180)}}} with {{{sqrt(180)}}} to get 180 (see scratch work above).




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More Scratch Work</font>


Following the same idea, {{{sqrt(5)*sqrt(180)=sqrt(5*180)}}} due to the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}. 


So  {{{sqrt(5)*sqrt(180)=sqrt(5*180)=sqrt(900)=sqrt(30^2)=30}}}


In the end, {{{sqrt(5)*sqrt(180)=30}}}



If you're not sure about your work, a calculator will come in handy to check your work.


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Now back to the problem at hand....



{{{30/180}}} Multiply {{{sqrt(5)}}} with {{{sqrt(180)}}} to get 30 (see scratch work above).



{{{1/6}}} Reduce the fraction.



So {{{sqrt(5)/sqrt(180)=1/6}}} regardless of the method used to get the answer.