Question

How do you simplify square roots? (examples of problems I need help with down below in the description)

Simplify,
1) `sqrt(20)`

2) `12/sqrt(3)`

3) `sqrt(6)/12`

Answer 1

See explanations below:

Explanation:

Example 1

The simplification here is to reduce the term within the radical to the smallest number possible using this rule. You factor the term inside of the radical to two terms. One which is a square, the other which is not:

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

`sqrt(20) => sqrt(color(red)(4) * color(blue)(5)) => sqrt(color(red)(4)) * sqrt(color(blue)(5)) => 2sqrt(5)`

Example 2

The simplification in this example is to rationalize the denominator. Or in other words, to remove all radicals from the denominator. You do this by multiplying by the appropriate form of `1`"

`12/sqrt(3) => sqrt(3)/sqrt(3) xx 12/sqrt(3) => (12sqrt(3))/(sqrt(3))^2 => (12sqrt(3))/3 => 4sqrt(3)`

Example 3

This is in simplified radical form. There denominator is rationalize - there is no radical in the denominator. The `sqrt(6)` cannot be simplified as in Example 1. We can put this into exponent form using this rule:

`root(color(red)(n))(x) = x^(1/color(red)(n))`

`sqrt(6)/12 => root(2)(6)/(2 * 6) => 6^(1/2)/(2 * 6^1) => 1/(2 * 6^(1-1/2)) => 1/(2 * 6^(1/2))`