How do you simplify #\root[ 5] { 19683x ^ { 7} y ^ { 14} }#?

1 Answer
George C.
Apr 8, 2017

#root(5)(19683x^7y^14) = 3xy^2root(5)(9x^2y^4)#

#root(7)(19683x^7y^14) = 3xy^2#

Explanation:

Factorise #19683#...

#color(white)(0000)19683#
#color(white)(0000)"/"color(white)(000)"\"#
#color(white)(000)3color(white)(000)6561#
#color(white)(0000000)"/"color(white)(00)"\"#
#color(white)(000000)3color(white)(000)2187#
#color(white)(0000000000)"/"color(white)(00)"\"#
#color(white)(000000000)3color(white)(000)729#
#color(white)(0000000000000)"/"color(white)(0)"\"#
#color(white)(000000000000)3color(white)(00)81#
#color(white)(00000000000000)"/"color(white)(00)"\"#
#color(white)(0000000000000)3color(white)(0000)9#
#color(white)(00000000000000000)"/"color(white)(0)"\"#
#color(white)(0000000000000000)3color(white)(000)3#

So:

#19683 = 3^7#

Hence:

#root(5)(19683x^7y^14) = root(5)(3^5x^5y^10 3^2x^2y^4)#

#color(white)(root(5)(19683x^7y^14)) = 3xy^2root(5)(9x^2y^4)#

Note that:

#root(7)(19683x^7y^14) = 3xy^2#

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