Question 29888
{{{ (root (3, 50x10y) * root(3, 25x16y))^7 }}}
I am assuming the whole thing here is to the 7th power, not just the second quantity.  firstly .. since they are of the same root, and you are multiplying, you can bring everything under one root.
{{{ (root(3, 50x*y*25x*16y))^7 }}}
prime factor all parts
{{{ (root(3, (5*5*2*x)*(5*2*y)*(5*5*x)*(2*2*2*2*y)))^7 }}}
rerite this so that all factors are grouped
{{{ (root(3, (5*5*5*5*5*2*2*2*2*2*2*x*x*y*y)))^7 }}}
since this is a cube root .... pull the factors out in groups of 3.
{{{ (root( 3, (highlight(5*5*5)*5*5*highlight(2*2*2)*highlight(2*2*2)*x*x*y*y)))^7 }}}
pull the 5, 2 and 2 out and multiply them
{{{ (20 (root( 3, (5*5*x*x*y*y))))^7 }}}
combine term within the root
{{{ (20 (root( 3, (25x^2y^2))))^7 }}}
now taking it piece by piece ....
{{{ 20^7 = 128 }}}
{{{ (root( 3, (25x^2y^2)))^7 }}} look at the root and the power ....
3 goes into 7 twice with one left over ...
{{{ (25x^2y^2) (25x^2y^2) (root( 3, (25x^2y^2))) }}} multiply by the 128
{{{ highlight (128 (25x^2y^2) (25x^2y^2)) (root( 3, (25x^2y^2))) }}}
{{{ (80000x^4y^4) (root( 3, (25x^2y^2))) }}}