How do you simplify #\frac { \root[ 3] { x ^ { 2} y ^ { 7} } } { \root [ 6] { x y ^ { 2} } }#?

3 Answers
Binayaka C.
Sep 8, 2017

# y^2 sqrtx #

Explanation:

#root(3)(x^2y^7)/root(6)(xy^2) #

#= (x^(2/3) * y^(7/3))/(x^(1/6) * y^(2/6)) #

#= x^(2/3 -1/6) * y^(7/3-1/3) #

#= x^(1/2) * y^2 = y^2sqrtx # [Ans]

sjc
Sep 9, 2017

#=y^2sqrtx#

Explanation:

Another approach is to put all the terms under the same root

#root(3)(x^2y^7)/(root(6)(xy^2))#

#=root(6)(x^4y^14)/(root(6)(xy^2)#

#root(6)[(x^4y^14)/(xy^2)]#

#=root(6)[x^3y^12)#

#=x^(3/6)y^(12/6)#

#=x^(1/2)y^(2)#

#=y^2sqrtx#

George C.
Sep 14, 2017

#root(3)(x^2y^7)/root(6)(xy^2) = sgn(y)*y^2sqrt(x)#

Explanation:

Given:

#root(3)(x^2y^7)/root(6)(xy^2)#

Assume #x# and #y# are real numbers with #x >= 0# and no other restriction on #y#.

Note that:

#root(3)(x^2y^7) = root(3)((y^2)^3)root(3)(x^2y) = y^2 root(3)(x^2y) = sgn(y)*y^2 root(6)(x^4 y^2)#

where:

#sgn(y) = { (1 " if " y > 0), (0 " if " y = 0), (-1 " if " y < 0) :}#

Then:

#root(3)(x^2y^7)/root(6)(xy^2) = (sgn(y)*y^2 root(6)(x^4 y^2))/root(6)(xy^2)#

#color(white)(root(3)(x^2y^7)/root(6)(xy^2)) = sgn(y)*y^2 root(6)((x^4 y^2)/(xy^2))#

#color(white)(root(3)(x^2y^7)/root(6)(xy^2)) = sgn(y)*y^2 root(6)(x^3)#

#color(white)(root(3)(x^2y^7)/root(6)(xy^2)) = sgn(y)*y^2 x^(3/6)#

#color(white)(root(3)(x^2y^7)/root(6)(xy^2)) = sgn(y)*y^2sqrt(x)#

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