Square root of 432 simplified.

Steps to Solve

1. Find the prime factorization of 432

Start by dividing 432 by the smallest prime numbers until you reach 1.

• Divide by 2: $$ 432 \div 2 = 216$$
• Divide by 2 again: $$ 216 \div 2 = 108$$
• Divide by 2 again: $$ 108 \div 2 = 54$$
• Divide by 2 again: $$ 54 \div 2 = 27$$
• Divide by 3 (the next smallest prime): $$ 27 \div 3 = 9$$
• Divide by 3 again: $$ 9 \div 3 = 3$$
• Divide by 3 again: $$ 3 \div 3 = 1$$

So, the prime factorization of 432 is: $$ 432 = 2^4 \times 3^3 $$

2. Apply the square root to the prime factorization

We can now apply the square root to each prime factor.

Using the property that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we get: $$ \sqrt{432} = \sqrt{2^4} \times \sqrt{3^3} $$

3. Simplify each square root

Now calculate the square roots:

• For $2^4$: $$ \sqrt{2^4} = 2^2 = 4 $$

• For $3^3$: $$ \sqrt{3^3} = \sqrt{3^2 \times 3} = 3 \times \sqrt{3} = 3 \sqrt{3} $$

4. Combine the results

Now, combine the simplified parts together: $$ \sqrt{432} = 4 \times 3 \sqrt{3} = 12 \sqrt{3} $$