Question 178266
First, let's simplify the numerator {{{sqrt(a-sqrt(a))*sqrt(a+sqrt(a))}}}


Let {{{x=sqrt(a)}}}


{{{sqrt(a-sqrt(a))*sqrt(a+sqrt(a))}}} Start with the given expression.



{{{sqrt(a-x)*sqrt(a+x)}}} Plug in {{{x=sqrt(a)}}}



{{{sqrt((a-x)(a+x))}}} Combine the roots using the identity  {{{sqrt(A)*sqrt(B)=sqrt(A*B)}}}



{{{sqrt(a^2-x^2)}}} FOIL



{{{sqrt(a^2-(sqrt(a))^2)}}} Plug in {{{x=sqrt(a)}}}



{{{sqrt(a^2-a)}}} Square {{{sqrt(a)}}} to get "a"



{{{sqrt(a(a-1))}}} Factor out the GCF "a"



{{{sqrt(a)*sqrt(a-1)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}} (this just the reverse of the previous identity)



So the numerator {{{sqrt(a-sqrt(a))*sqrt(a+sqrt(a))}}} simplifies to {{{sqrt(a)*sqrt(a-1)}}} 

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Now let's go back to the main problem:



So the expression



{{{(sqrt(a-sqrt(a))*sqrt(a+sqrt(a)))/(sqrt(a-1))}}}



simplifies to 



{{{(sqrt(a)*sqrt(a-1))/(sqrt(a-1))}}} (see steps above)




{{{(sqrt(a)*highlight(sqrt(a-1)))/(highlight(sqrt(a-1)))}}} Highlight the common terms.



{{{(sqrt(a)*cross(sqrt(a-1)))/(cross(sqrt(a-1)))}}} Cancel out the common terms.



{{{sqrt(a)}}} Simplify



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Answer:



So {{{(sqrt(a-sqrt(a))*sqrt(a+sqrt(a)))/(sqrt(a-1))=sqrt(a)}}} where {{{a>1}}}