How do you simplify #4sqrt (81 / 8)#?

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Answer 1 · 2017-06-22T21:16:10

See a solution process below:

First, using this rule of radicals rewrite the expression:

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

#4sqrt(color(red)(81)/color(blue)(8)) => 4sqrt(color(red)(81))/sqrt(color(blue)(8)) => (4 * 9)/sqrt(8) => 36/sqrt(8)#

Next, rewrite the denominator and use this rule of exponents to simplify the denominator:

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

#36/sqrt(8) => 36/sqrt(color(red)(4) * color(blue)(2)) => 36/(sqrt(color(red)(4)) * sqrt(color(blue)(2))) => 36/(2sqrt(2)) => 18/sqrt(2)#

To rationalize the denominator or eliminate the radical from the denominator we can multiply by the appropriate form of #1#:

#18/sqrt(2) xx sqrt(2)/sqrt(2) => (18sqrt(2))/(sqrt(2))^2 => (18sqrt(2))/2 => 9sqrt(2)#

Answer 2 · 2017-06-22T22:51:26

#4sqrt(81/8) = 9sqrt(2)#

If #a, b > 0# then:

#sqrt(a/b) = sqrt(a)/sqrt(b)#

When simplifying square roots of fractions such as this example, I like to make the denominator into a perfect square first.

So we find:

#4sqrt(81/8) = 4sqrt((81*2)/16) = 4sqrt((9^2*2)/4^2) = 4*9/4sqrt(2) = 9sqrt(2)#