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What is the square root of 11664?

Understand the Problem

The question is asking for the square root of the number 11664. To solve this, we will look for a number which, when multiplied by itself, gives us 11664.

Answer

The square root of 11664 is $54$.
Answer for screen readers

The square root of 11664 is $54$.

Steps to Solve

  1. Identify the Problem We need to find the square root of the number 11664, which is represented by the equation:

$$ \sqrt{11664} = x $$

where $x$ is the number we are trying to find.

  1. Use Prime Factorization We can find the square root by prime factorization. First, we need to factorize 11664 into its prime factors.

After performing prime factorization, we will get:

$$ 11664 = 2^2 \times 3^6 $$

  1. Apply the Square Root Rule Use the property of square roots that states:

$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$

We can apply this to our prime factorization:

$$ \sqrt{11664} = \sqrt{2^2 \times 3^6} = \sqrt{2^2} \times \sqrt{3^6} $$

  1. Calculate Each Square Root Now, calculate the square roots of each factor:
  • For $2^2$:

$$ \sqrt{2^2} = 2 $$

  • For $3^6$:

$$ \sqrt{3^6} = 3^{6/2} = 3^3 = 27 $$

  1. Combine the Results Now, multiply the results from the previous step:

$$ \sqrt{11664} = 2 \times 27 = 54 $$

The square root of 11664 is $54$.

More Information

The square root of a number is a value that, when multiplied by itself, gives that number. In this case, $54 \times 54 = 11664$. This problem also demonstrates the power of prime factorization in simplifying calculations.

Tips

  • Forgetting to use prime factorization properly can lead to an incorrect answer. Make sure to correctly factor the number into its constituent primes.
  • Not applying the square root property accurately for each factor.
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