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
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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 $x = \sqrt{2592}$.
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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: $$ 2592 = 2^5 \times 3^4 $$
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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: $$ \sqrt{2592} = 4\sqrt{2} \times 9 = 36\sqrt{2} $$
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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.
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