Question 212370
I'll do the first one to get you going in the right direction.


# 1



{{{4*sqrt(405x^8y^10)}}} Start with the given expression.



{{{4*sqrt(405)*sqrt(x^8)*sqrt(y^10)}}} Break up the square root using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}



{{{4*sqrt(81*5)*sqrt(x^8)*sqrt(y^10)}}} Factor 405 into 81*5



{{{4*sqrt(9^2*5)*sqrt(x^8)*sqrt(y^10)}}} Rewrite 81 as {{{9^2}}}



{{{4*sqrt(9^2*5)*sqrt((x^4)^2)*sqrt(y^10)}}} Rewrite {{{x^8}}} as {{{(x^4)^2}}}



{{{4*sqrt(9^2*5)*sqrt((x^4)^2)*sqrt((y^5)^2)}}} Rewrite {{{y^10}}} as {{{(y^5)^2}}}



{{{4*sqrt(9^2)*sqrt(5)*sqrt((x^4)^2)*sqrt((y^5)^2)}}} Break up the first square root (using that same identity)



{{{4*9*sqrt(5)*sqrt((x^4)^2)*sqrt((y^5)^2)}}} Take the square root of {{{9^2}}} to get 9 (notice how the squares and the square roots 'undo' each other).



{{{4*9*sqrt(5)*x^4*sqrt((y^5)^2)}}} Take the square root of {{{(x^4)^2}}} to get {{{x^4}}}.



{{{4*9*sqrt(5)*x^4*y^5}}} Take the square root of {{{(y^5)^2}}} to get {{{y^5}}}.



{{{36x^4y^5*sqrt(5)}}} Rearrange the terms.



So {{{4*sqrt(405x^8y^10)=36x^4y^5*sqrt(5)}}} where every variable is nonnegative.