What is the square root of 2592?

Steps to Solve

1. Determine the square root operation To find the square root of a number, we are looking for a number $x$ such that $x^2 = 2592$. This means we want to find $x = \sqrt{2592}$.

2. Factor the number 2592 We will break down 2592 into its prime factors.

Starting by dividing 2592 by 2 repeatedly: $$ 2592 \div 2 = 1296 $$ $$ 1296 \div 2 = 648 $$ $$ 648 \div 2 = 324 $$ $$ 324 \div 2 = 162 $$ $$ 162 \div 2 = 81 $$ At this point, 81 is not divisible by 2, so we switch to the next prime number, which is 3: $$ 81 \div 3 = 27 $$ $$ 27 \div 3 = 9 $$ $$ 9 \div 3 = 3 $$ $$ 3 \div 3 = 1 $$ Now, we have the complete factorization: $$ 2592 = 2^5 \times 3^4 $$

3. Use the prime factorization to find the square root The square root of a product of prime factors can be found by taking half the power of each prime: $$ \sqrt{2592} = \sqrt{2^5 \times 3^4} = \sqrt{2^5} \times \sqrt{3^4} $$ Calculating each part: $$ \sqrt{2^5} = 2^{\frac{5}{2}} = 2^2 \times \sqrt{2} = 4\sqrt{2} $$ $$ \sqrt{3^4} = 3^{\frac{4}{2}} = 3^2 = 9 $$ Now combining them: $$ \sqrt{2592} = 4\sqrt{2} \times 9 = 36\sqrt{2} $$

4. Approximate the final answer To find a numerical approximation, we can calculate $\sqrt{2} \approx 1.414$: $$ 36\sqrt{2} \approx 36 \times 1.414 \approx 50.904 $$