Simplify the square root of 192.

Steps to Solve

1. Factor the number 192 into prime factors

First, we need to find the prime factors of 192. We can do this by dividing by the smallest prime numbers.

Starting with 192: $$ 192 \div 2 = 96 \ 96 \div 2 = 48 \ 48 \div 2 = 24 \ 24 \div 2 = 12 \ 12 \div 2 = 6 \ 6 \div 2 = 3 \ 3 \div 3 = 1 $$

So, the prime factorization of 192 is $2^6 \times 3^1$.

2. Apply the square root to the prime factors

Next, we will apply the square root to the prime factorization. The square root of a product can be expressed as the product of the square roots:

$$ \sqrt{192} = \sqrt{2^6 \times 3^1} = \sqrt{2^6} \times \sqrt{3^1} $$

3. Simplify the square roots

We know that $\sqrt{a^b} = a^{b/2}$. Thus:

$$ \sqrt{2^6} = 2^{6/2} = 2^3 = 8 $$

For $3^1$:

$$ \sqrt{3^1} = \sqrt{3} $$

Combine them:

$$ \sqrt{192} = 8 \times \sqrt{3} $$

4. Write the final simplified form

Therefore, the simplified form of $\sqrt{192}$ is:

$$ \sqrt{192} = 8\sqrt{3} $$