square root of 72 simplified radical form

Steps to Solve

1. Prime Factorization of 72

To simplify the square root, we first need to find the prime factors of 72.

Starting with the smallest prime number, 2:

$$ 72 \div 2 = 36 $$

$$ 36 \div 2 = 18 $$

$$ 18 \div 2 = 9 $$

Next, switch to the next prime number, which is 3:

$$ 9 \div 3 = 3 $$

$$ 3 \div 3 = 1 $$

So, the prime factorization of 72 is:

$$ 72 = 2^3 \times 3^2 $$

2. Apply the Square Root

Now we apply the square root to the prime factorization:

$$ \sqrt{72} = \sqrt{2^3 \times 3^2} $$

3. Separate the Square Roots

We can separate the square roots of the factors:

$$ \sqrt{72} = \sqrt{2^3} \times \sqrt{3^2} $$

4. Simplify Each Square Root

Next, we simplify each square root:

• For $\sqrt{3^2}$, this simplifies to $3$.
• For $\sqrt{2^3}$, we can split it as $\sqrt{2^2 \times 2}$, which simplifies to $2\sqrt{2}$.

So, we have:

$$ \sqrt{2^3} = 2\sqrt{2} $$

Thus:

$$ \sqrt{72} = 2\sqrt{2} \times 3 $$

5. Combine the Results

Now, we multiply the simplified parts together:

$$ 2\sqrt{2} \times 3 = 6\sqrt{2} $$

So, the simplified radical form of $\sqrt{72}$ is:

$$ \sqrt{72} = 6\sqrt{2} $$