Question

How do you simplify `\frac { 5\sqrt { 5a ^ { 2} } } { 3\sqrt { 3a ^ { 4} } }`?

Answer 1

See a solution process below:

Explanation:

First, rewrite the expression as:

`5/3(sqrt(5a^2)/sqrt(3a^4))`

Next, use this rule for radicals to again rewrite the expression:

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

`5/3(sqrt(color(red)(5a^2))/sqrt(color(blue)(3a^4))) => 5/3sqrt(color(red)(5a^2)/color(blue)(3a^4)) => 5/3sqrt(color(red)(5)/color(blue)(3a^2))`

Next, use this rule to rewrite and simplify the expression:

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

`5/3sqrt(color(red)(5)/color(blue)(3a^2)) => 5/3sqrt(color(red)(5)/color(blue)(3) *1/a^2) => 5/3sqrt(color(red)(5)/color(blue)(3))sqrt(1/a^2) =>`

`5/3sqrt(5/3)1/a`

If necessary we can rewrite the radical as:

`5/3(5/3)^(1/2)1/a => (5/3)^1(5/3)^(1/2)1/a =>`

`(5/3)^(2/2)(5/3)^(1/2)1/a => (5/3)^(2/2+1/2)1/a => (5/3)^(3/2)1/a`