Question 195463
# 1



{{{sqrt(54x^4)}}} Start with the given expression



{{{sqrt(9*6x^4)}}} Factor 54 into {{{9*6}}}. Note: 9 is the largest perfect square factor of 54



{{{sqrt(9*6*x^2*x^2)}}} Factor {{{x^4}}} into {{{x^2*x^2}}}



{{{sqrt(9)*sqrt(6)*sqrt(x^2)*sqrt(x^2)}}} Break up the square root.



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



{{{3*sqrt(6)*x*x}}} Take the square root of {{{x^2}}} to get "x"



{{{3x*sqrt(6)}}} Multiply



So {{{sqrt(54x^4)=3x*sqrt(6)}}} where {{{x>=0}}}



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# 2



{{{6w*sqrt(63*u^3)-u*sqrt(7*uw^2)}}} Start with the given expression



{{{6w*3u*sqrt(7u)-u*sqrt(7*uw^2)}}} Simplify {{{sqrt(63*u^3)}}} to get {{{3*u*sqrt(7*u)}}}



{{{6w*3u*sqrt(7u)-uw*sqrt(7u)}}} Simplify {{{sqrt(7*uw^2)}}} to get {{{w*sqrt(7*u)}}}



{{{18uw*sqrt(7u)-uw*sqrt(7u)}}} Multiply



{{{(18-1)uw*sqrt(7u)}}} Factor out the GCF {{{uw*sqrt(7u)}}}



{{{17uw*sqrt(7u)}}} Combine like terms.




So {{{6w*sqrt(63*u^3)-u*sqrt(7*uw^2)=17uw*sqrt(7u)}}} where every variable is non negative.



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# 3


{{{sqrt(21)/sqrt(77)}}} Start with the given expression



{{{(sqrt(21)sqrt(77))/(sqrt(77)sqrt(77))}}} Multiply both the numerator and denominator by {{{sqrt(77)}}}



{{{(sqrt(21)sqrt(77))/(77)}}} Multiply {{{sqrt(77)sqrt(77)}}} to get {{{sqrt(77)sqrt(77)=(sqrt(77))^2=77}}}



{{{(7*sqrt(33))/(77)}}} Multiply {{{sqrt(21)sqrt(77)}}} to get {{{sqrt(21)sqrt(77)=sqrt(21*77)=sqrt(7*3*7*11)=sqrt(49*33)=7*sqrt(33)}}}



{{{sqrt(33)/11}}} Reduce



So {{{sqrt(21)/sqrt(77)=sqrt(33)/11}}}