How do you simplify #\sqrt { \frac { x ^ { 9} z ^ { 2} } { 18y ^ { 5} } }#?

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Answer 1 · 2017-07-16T01:05:37

See a solution process below:

First, rewrite this expression in two steps:

#sqrt(x^9z^2)/sqrt(18y^5) => sqrt(x^8z^2 * x)/sqrt(9y^4 * 2y)#

Next, use this rule for radicals to simplify the radicals:

#sqrt(color(red)(a) * color(blue)(b)) = sqrt(color(red)(a)) * sqrt(color(blue)(b))#

#sqrt(x^8z^2 * x)/sqrt(9y^4 * 2y) =>#

#(sqrt(x^8z^2)sqrt(x))/(sqrt(9y^4)sqrt(2y)) =>#

#(x^4z^1sqrt(x))/(3y^2sqrt(2y))#

One way we can rewrite this is:

#(x^4z^1)/(3y^2)sqrt(x/(2y))#

Another way is to rationalize the denominator. This means to remove all the radicals from the denominator:

#(x^4z^1sqrt(x))/(3y^2sqrt(2y)) xx sqrt(2y)/sqrt(2y) =>#

#(x^4z^1sqrt(x) xx sqrt(2y))/(3y^2sqrt(2y) xx sqrt(2y)) =>#

#(x^4z^1sqrt(x xx 2y))/(3y^2 * 2y) =>#

#(x^4z^1sqrt(2xy))/(6y^3)#