Question 189060
{{{sqrt((36a^9b^12)/(3ab^7))}}} Start with the given expression.



{{{sqrt((12a^9b^12)/(ab^7))}}} Divide 36 into 3 to get 12



{{{sqrt(12a^(9-1)b^(12-7))}}} Divide the variable terms by subtracting the corresponding exponents.



{{{sqrt(12a^8b^5)}}} Subtract



{{{sqrt(4*3*a^8*b^5)}}} Factor {{{12}}} into {{{4*3}}}



{{{sqrt(4*3*a^2*a^2*a^2*a^2*b^5)}}} Factor {{{a^8}}} into {{{a^2*a^2*a^2*a^2}}}



{{{sqrt(4*3*a^2*a^2*a^2*a^2*b^2*b^2*b)}}} Factor {{{b^5}}} into {{{b^2*b^2*b}}}



{{{sqrt(4)*sqrt(3)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}}.



{{{2*sqrt(3)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b)}}} Take the square root of {{{4}}} to get {{{2}}}.



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



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



{{{2a^4b^2*sqrt(3b)}}} Rearrange and multiply the terms.


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Answer:



So {{{sqrt((36a^9b^12)/(3ab^7))}}} simplifies to {{{2a^4b^2*sqrt(3b)}}}



In other words, {{{sqrt((36a^9b^12)/(3ab^7))=2a^4b^2*sqrt(3b)}}} where {{{a>0}}} and {{{b>0}}}