What is the square root of 1764?

Steps to Solve

1. Identify the Problem We need to find the square root of 1764, denoted as $\sqrt{1764}$. This means we are looking for a number $x$ such that $x^2 = 1764$.

2. Prime Factorization To solve for the square root, we can factor 1764 into its prime factors. This helps in simplifying the square root:

First, we can divide 1764 by 2 (the smallest prime number): $$ 1764 \div 2 = 882 $$ Next, divide 882 by 2: $$ 882 \div 2 = 441 $$ Now, 441 is not divisible by 2, so we move to the next prime number, which is 3: $$ 441 \div 3 = 147 $$ Divide 147 by 3: $$ 147 \div 3 = 49 $$ Finally, we recognize that 49 is $7^2$: $$ 49 = 7 \times 7 $$ Putting it all together, the prime factorization of 1764 is: $$ 1764 = 2^2 \times 3^2 \times 7^2 $$

3. Apply the Square Root Property Using the property of square roots, where $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can find the square root of 1764: $$ \sqrt{1764} = \sqrt{2^2 \times 3^2 \times 7^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{7^2} $$

4. Calculate the Square Root Now we can calculate: $$ \sqrt{2^2} = 2, \quad \sqrt{3^2} = 3, \quad \sqrt{7^2} = 7 $$ Therefore, $$ \sqrt{1764} = 2 \times 3 \times 7 $$

5. Final Multiplication Now, we perform the final multiplication: $$ 2 \times 3 = 6 $$ Then: $$ 6 \times 7 = 42 $$