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
- 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.
- 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:
$$ 11664 = 2^2 \times 3^6 $$
- 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:
$$ \sqrt{11664} = \sqrt{2^2 \times 3^6} = \sqrt{2^2} \times \sqrt{3^6} $$
- Calculate Each Square Root Now, calculate the square roots of each factor:
- For $2^2$:
$$ \sqrt{2^2} = 2 $$
- For $3^6$:
$$ \sqrt{3^6} = 3^{6/2} = 3^3 = 27 $$
- 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.
