What is the cube root of 46656?
Understand the Problem
The question is asking for the cube root of the number 46656. To find the cube root, we are looking for a number that, when multiplied by itself three times, equals 46656.
Answer
$36$
Answer for screen readers
The cube root of 46656 is 36.
Steps to Solve
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Recognizing the operation We need to find the cube root of 46656. This is represented mathematically as $x = \sqrt[3]{46656}$.
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Using prime factorization To find the cube root, we can first break down 46656 into its prime factors. Upon dividing repeatedly, we can find that: $$ 46656 = 2^6 \times 3^6 $$
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Applying the property of cube roots To calculate the cube root, we can use the property of exponents: $$ \sqrt[3]{a^m} = a^{m/3} $$ So, using the prime factorization we found: $$ \sqrt[3]{46656} = \sqrt[3]{(2^6 \times 3^6)} = 2^{6/3} \times 3^{6/3} = 2^2 \times 3^2 $$
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Calculating the final result Now we can calculate this: $$ 2^2 = 4 \quad \text{and} \quad 3^2 = 9 $$ Thus, multiplying these results: $$ 4 \times 9 = 36 $$
The cube root of 46656 is 36.
More Information
The cube root of a number is the value that, when multiplied by itself three times, gives the original number. In the case of 46656, the prime factorization method demonstrates how we can break down the number for easier calculation.
Tips
- Forgetting to check for prime factors can lead to errors in determining the cube root.
- Misapplying the cube root property could result in incorrect simplifications.
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