What is the square root of 2592?

Steps to Solve

1. Identify the square root operation To find the square root of a number, we need to find a value that, when multiplied by itself, gives us that number. In this case, we are looking for $\sqrt{2592}$.

2. Prime factorization of 2592 We can simplify the square root by finding the prime factorization of 2592. We can divide by the smallest prime number, 2, until we reach a prime number.

   • $2592 \div 2 = 1296$
   • $1296 \div 2 = 648$
   • $648 \div 2 = 324$
   • $324 \div 2 = 162$
   • $162 \div 2 = 81$
   • $81 \div 3 = 27$
   • $27 \div 3 = 9$
   • $9 \div 3 = 3$
   • $3 \div 3 = 1$

This gives us the prime factorization: $2592 = 2^5 \times 3^4$.

3. Applying the square root to the prime factorization Using the prime factorization, we can find the square root:
   • The square root of $2^5$ is $2^{\frac{5}{2}} = 2^2 \times \sqrt{2} = 4\sqrt{2}$.
   • The square root of $3^4$ is $3^{\frac{4}{2}} = 3^2 = 9$.

Now we can combine these together: $$ \sqrt{2592} = \sqrt{2^5 \times 3^4} = 4\sqrt{2} \times 9. $$

4. Simplifying the solution Now multiply the coefficients: $$ 4 \times 9 = 36. $$ Thus, we have: $$ \sqrt{2592} = 36\sqrt{2}. $$

5. Final Value The final answer for the square root of 2592 is $36\sqrt{2}$.