What is the square root of 648?

Steps to Solve

1. Prime Factorization of 648

To calculate the square root of 648, we first find its prime factorization.

648 can be divided as follows:

• $648 \div 2 = 324$
• $324 \div 2 = 162$
• $162 \div 2 = 81$
• $81 \div 3 = 27$
• $27 \div 3 = 9$
• $9 \div 3 = 3$
• $3 \div 3 = 1$

So, the prime factorization of 648 is: $$648 = 2^3 \times 3^4$$

2. Grouping the Factors

Next, we group the factors in pairs to find the square root. In square roots, a pair of the same factor will come out of the root as a single factor.

From the prime factorization:

• For $2^3$, we can take out one pair of $2$ and leave one $2$ inside the root.
• For $3^4$, we can take out two pairs of $3$.

Thus: $$\sqrt{648} = \sqrt{2^3 \times 3^4} = \sqrt{(2^2 \times 3^4) \times 2} = 2^1 \times 3^2 \times \sqrt{2}$$

3. Calculating the Square Root

We now simplify the expression:

• Calculate $2^1 = 2$
• Calculate $3^2 = 9$

So: $$\sqrt{648} = 2 \times 9 \times \sqrt{2} = 18\sqrt{2}$$