Question 187246

{{{sqrt(54*a^5*b^7)}}} Start with the given expression.



{{{sqrt(9*6*a^5*b^7)}}} Factor {{{54}}} into {{{9*6}}}



{{{sqrt(9*6*a^2*a^2*a*b^7)}}} Factor {{{a^5}}} into {{{a^2*a^2*a}}}



{{{sqrt(9*6*a^2*a^2*a*b^2*b^2*b^2*b)}}} Factor {{{b^7}}} into {{{b^2*b^2*b^2*b}}}



{{{sqrt(9)*sqrt(6)*sqrt(a^2)*sqrt(a^2)*sqrt(a)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}}.



{{{3*sqrt(6)*sqrt(a^2)*sqrt(a^2)*sqrt(a)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b)}}} Take the square root of {{{9}}} to get {{{3}}}.



{{{3*sqrt(6)*a*a*sqrt(a)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b)}}} Take the square root of {{{a^2}}} to get {{{a}}}.



{{{3*sqrt(6)*a*a*sqrt(a)*b*b*b*sqrt(b)}}} Take the square root of {{{b^2}}} to get {{{b}}}.



{{{3a^2b^3*sqrt(6ab)}}} Rearrange and combine the terms.


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Answer:



So {{{sqrt(54*a^5*b^7)}}} simplifies to {{{3a^2b^3*sqrt(6ab)}}}



In other words, {{{sqrt(54*a^5*b^7)=3a^2b^3*sqrt(6ab)}}} where every variable is nonnegative.