Find the cube root of 46656.

Answer

To find the cube root of 46656, we will follow these steps:

### Step 1: Prime Factorization
We start by performing the prime factorization of 46656.

1. Divide 46656 by 2:
- 46656 ÷ 2 = 23328
2. Divide 23328 by 2:
- 23328 ÷ 2 = 11664
3. Divide 11664 by 2:
- 11664 ÷ 2 = 5832
4. Divide 5832 by 2:
- 5832 ÷ 2 = 2916
5. Divide 2916 by 2:
- 2916 ÷ 2 = 1458
6. Divide 1458 by 2:
- 1458 ÷ 2 = 729

Now, we can no longer divide by 2, so we switch to the next prime number, which is 3.

7. Divide 729 by 3:
- 729 ÷ 3 = 243
8. Divide 243 by 3:
- 243 ÷ 3 = 81
9. Divide 81 by 3:
- 81 ÷ 3 = 27
10. Divide 27 by 3:
- 27 ÷ 3 = 9
11. Divide 9 by 3:
- 9 ÷ 3 = 3
12. Divide 3 by 3:
- 3 ÷ 3 = 1

So, the prime factorization of 46656 is:
\[ 46656 = 2^6 \times 3^6 \]

### Step 2: Finding the Cube Root
Now, we need to find the cube root of 46656, which can be expressed as:
\[ \sqrt[3]{46656} = \sqrt[3]{2^6 \times 3^6} \]

Using the property of cube roots:
\[ \sqrt[3]{a^m} = a^{m/3} \]

We can rewrite the expression:
\[ \sqrt[3]{2^6 \times 3^6} = \sqrt[3]{2^6} \times \sqrt[3]{3^6} \]

Calculating each part:
1. \( \sqrt[3]{2^6} = 2^{6/3} = 2^2 = 4 \)
2. \( \sqrt[3]{3^6} = 3^{6/3} = 3^2 = 9 \)

### Step 3: Multiply the Results
Now, we multiply the results from the cube roots:
\[ 4 \times 9 = 36 \]

### Final Answer
Thus, the cube root of 46656 is:
\[ \sqrt[3]{46656} = 36 \]

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