How do you simplify #\frac { \sqrt { 72x ^ { 3} } } { \sqrt { 9x ^ { - 5} } }#?

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Answer 1 · 2017-06-10T18:15:04

See a solution process below:

First, we can use this rule of radicals to rewrite this expression:

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

#sqrt(color(red)(72x^3))/sqrt(color(blue)(9x^-5)) = sqrt(color(red)(72x^3)/color(blue)(9x^-5)) => sqrt((8color(red)(x^3))/color(blue)(x^-5))#

We can now use this rule of exponents to simplify the #x# terms within the radical:

#x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))#

#sqrt((8color(red)(x^3))/color(blue)(x^-5)) => sqrt( 8color(red)(x^(3-color(blue)(-5)))) => sqrt(8color(red)(x^(3+color(blue)(5)))) => sqrt(8x^8)#

We can now use this rule for radicals to simplify the remain expression:

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

#sqrt(8x^8) => sqrt(color(red)(4x^8) * color(blue)(2)) => sqrt(color(red)(4x^8)) * sqrt(color(blue)(2)) => 2x^4sqrt(2)#