What is the square root of 46656?
Understand the Problem
The question is asking for the square root of the number 46656, which requires calculating the value that, when multiplied by itself, equals 46656.
Answer
$216$
Answer for screen readers
The square root of 46656 is $216$.
Steps to Solve
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Expressing the Problem We need to find a number $x$ such that $x^2 = 46656$. Therefore, we are calculating the square root: $x = \sqrt{46656}$.
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Finding Prime Factorization To simplify this, we can perform the prime factorization of 46656. Starting by dividing by the smallest prime numbers: $$ 46656 \div 2 = 23328 \ 23328 \div 2 = 11664 \ 11664 \div 2 = 5832 \ 5832 \div 2 = 2916 \ 2916 \div 2 = 1458 \ 1458 \div 2 = 729 \ 729 \div 3 = 243 \ 243 \div 3 = 81 \ 81 \div 3 = 27 \ 27 \div 3 = 9 \ 9 \div 3 = 3 \ 3 \div 3 = 1 $$ Now we write the prime factorization: $$ 46656 = 2^6 \cdot 3^6 $$
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Using the Square Root of Factorization The square root of a number can be found by taking the square root of its prime factorization: $$ \sqrt{46656} = \sqrt{2^6 \cdot 3^6} = \sqrt{(2^3)^2 \cdot (3^3)^2} = 2^3 \cdot 3^3 $$
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Calculating the Final Result Now we calculate $2^3$ and $3^3$: $$ 2^3 = 8 \ 3^3 = 27 $$ Therefore: $$ \sqrt{46656} = 8 \cdot 27 = 216 $$
The square root of 46656 is $216$.
More Information
The number 216 is a perfect cube as well, since $6^3 = 216$. This means that the number can also be expressed as the cube of 6.
Tips
- Forgetting to perform prime factorization step correctly can lead to an incorrect square root.
- Confusing square roots with cube roots might cause additional confusion.
