What is the Square Root of 252 in Simplest Radical Form?

What is the Square Root of 252 in Simplest Radical Form?

The square root of 252 in simplest radical form is √(36 * 7), which can be further simplified as √36 * √7, resulting in 6√7. Therefore, √252 simplified in simplest radical form is 6√7.

Finding Square Roots

In mathematics, the square root of a number n is defined as the number that, when multiplied by itself, equals n. It is represented by the radical symbol √n.

For example, √9 = 3 because 3 x 3 = 9. So 3 is the square root of 9.

Square roots are the reverse operation of squaring a number. Squaring means multiplying a number by itself.

To find the square root of a number by hand:

1. Break the number into prime factors
2. Pair up identical prime factors into multiples
3. Take one factor from each pair as the square root

Let’s find the square root of 252 using this process.

Prime Factoring 252

252 = 2 x 2 x 3 x 3 x 7

Notice 252 is already expressed as its prime factors. If it was not, we would need to factor it first before proceeding.

Pairing Identical Factors

252 = (2 x 2) x (3 x 3) x 7

We paired the twos together and the threes together since they are identical primes.

Taking One From Each Pair

√252 = 2 x 3 x √7

This gives us the simplified square root of 252.

Radical Form

The square root of a number cannot be further simplified if it contains a factor that is already prime (like 7).

So the square root of 252 expressed in simplest radical form is:

√252 = 2√7 x 3

Checking the Answer

We can verify this is correct by squaring the simplified square root to see if it gives the original number:

(2√7 x 3)2 = 252

Since squaring the square root returns 252, this confirms that 2√7 x 3 is the simplified radical form of √252.

Understanding prime factorization allows us to reverse the process to find square roots. Practicing finding roots by hand builds algebraic skills and number sense.

In summary, take the square root of a number by factoring into primes, pairing identical factors, and taking one factor from each pair to form the simplified radical.