What is the square root of 2592?
Understand the Problem
The question is asking for the square root of the number 2592, which is a mathematical operation where we need to find a number that, when multiplied by itself, equals 2592.
Answer
The square root of 2592 is $36\sqrt{2}$ or approximately $50.904$.
Answer for screen readers
The square root of 2592 is approximately $36\sqrt{2}$ or $50.904$.
Steps to Solve

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 $x = \sqrt{2592}$.

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: $$ 2592 = 2^5 \times 3^4 $$

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: $$ \sqrt{2592} = 4\sqrt{2} \times 9 = 36\sqrt{2} $$

Approximate the final answer To find a numerical approximation, we can calculate $\sqrt{2} \approx 1.414$: $$ 36\sqrt{2} \approx 36 \times 1.414 \approx 50.904 $$
The square root of 2592 is approximately $36\sqrt{2}$ or $50.904$.
More Information
This problem involves the concept of breaking down numbers into their prime factors, which is a useful technique in various areas of mathematics, including simplifying square roots and understanding number properties. Moreover, $36\sqrt{2}$ is an exact form, while $50.904$ is an approximate decimal representation.
Tips
 Not fully factoring the number correctly may lead to an incorrect square root.
 Confusing square roots with other operations, such as squaring, can result in errors.
 Failing to simplify the radical form to its simplest expression may also occur.