Question 118228
{{{sqrt(144*x^10y^12z^18)}}} Start with the given expression

{{{sqrt(144*x^2*x^2*x^2*x^2*x^2*y^2*y^2*y^2*y^2*y^2*y^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2)}}} Factor {{{x^10y^12z^18}}} into {{{x^2*x^2*x^2*x^2*x^2*y^2*y^2*y^2*y^2*y^2*y^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2*z^2}}}
 
{{{sqrt(144)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Break up the square root using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}
 
{{{12*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Take the square root of the perfect square 144 to get 12
 
{{{12*x*x*x*x*x*y*y*y*y*y*y*z*z*z*z*z*z*z*z*z}}} Take the square root of the perfect squares {{{x^2}}}, {{{y^2}}} and {{{z^2}}} to get {{{x}}}, {{{y}}} and {{{z}}} 
 
{{{12*x^5y^6z^9}}} Multiply the common terms 




So {{{sqrt(144*x^10y^12z^18)}}} simplifies to {{{12*x^5y^6z^9}}}




-------------------------------------


Here's another way to do it



{{{sqrt(144*x^10y^12z^18)}}} Start with the given expression



{{{(144*x^10y^12z^18)^(1/2)}}} Rewrite the radical expression into an exponential expression



{{{((144^1)*x^10y^12z^18)^(1/2)}}} Rewrite 144 as {{{144^1}}}



{{{144^(1*(1/2))*x^(10*(1/2))y^(12*(1/2))z^(18*(1/2))}}} Distribute the outer exponent



{{{144^(1/2)*x^(10/2)y^(12/2)z^(18/2)}}} Multiply the exponents



{{{144^(1/2)*x^5y^6z^9}}} Simplify



{{{sqrt(144)*x^5y^6z^9}}} Rewrite {{{144^(1/2)}}} as {{{sqrt(144)}}}



{{{12*x^5y^6z^9}}} Take the square root of the perfect square 144 to get 12



So {{{sqrt(144*x^10y^12z^18)}}} simplifies to {{{12*x^5y^6z^9}}}