How do you find the square root of 405?

To find the square root of 405, we can use the following detailed steps:

Prime factorization of 405:
Begin by breaking 405 into its prime factors:
405 ÷ 5 = 81
81 = 3 × 3 × 3 × 3
So, the prime factorization of 405 is:
405 = 3 × 3 × 3 × 3 × 5

Pair the prime factors:
For square roots, we take one factor from each pair:

Pair of 3s: Take one 3
Another pair of 3s: Take another 3
So, √(3 × 3 × 3 × 3 × 5) = 3 × 3 × √5 = 9√5.

Approximation of √5:
The value of √5 is approximately 2.236.
Substituting this value:
9 × 2.236 = 20.124.

Thus, the square root of 405 is approximately 20.124. For exact results, it is expressed as
9
√
5
9√5.