When factoring radicals, the following formula is useful, important, and a must use:

√(a × b) = √(a) × √(b)

Basically, the formula tells you that the square root of the multiplications is the same thing as the multiplication of the square roots

Let's not spend too much time on definition and English here. I am no English major. Let's start with examples

Once you know how to factor radicals, simplifying radicals(the ultimate goal) is very straightforward

Factor √(24)

24 = 12 × 2

24 = 4 × 6

24 = 3 × 8

Therefore, √(24) can be factored 3 different ways. Using the formula above, we get:

√(24) = √(2 × 12) = √(2) × √(12)

√(24) = √(4 × 6) = √(4) × √(6)

√(24) = √(3 × 8) = √(3) × √(8)

If you try to simplify using the equation 24 = 2 × 12, you will get the same answer as above

√(24)= √(12 × 2)= √(4 × 3 × 2) = √( 4 × 6) = √(4) × √(6)= 2 × √(6)

Factor and simplify √(72)

72 = 9 × 8

72 = 36 × 2

72 = 3 × 24

72 = 18 × 4

72 = 12 × 6

A lot more choices than before! So factoring radicals can quickly become a lot of work

√(72) = √(9 × 8) = √(9) × √(8)

√(72) = √(36 × 2) = √(36) × √(2)

√(72) = √(3 × 24) = √(3) × √(24)

√(72) = √(18 × 4) = √(18) × √(4)

√(72) = √(12 × 6) = √(12) × √(6)

To simplify now, we can use √(72) = √(9 × 8) = √(9) × √(8)

√(72) = √(9 × 8) = √(9) × √(8) = 3 × √ (8) = 3 × √ (4 × 2) = 3 × √(4) × √(2) = 3 × 2 × √ (2) = 6 × √ (2)

The problem can be simplified faster using √(72) = √(36 × 2) = √(36) × √(2) = 6 × √ (2)