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

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)`

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)`