To find the cube root of 46656 using the prime factorization method, we need to follow these steps:

Step 1: Prime Factorization of 46656

We start by dividing 46656 by the smallest prime number (2) and continue dividing until we reach a prime number.

$46656 \div 2 = 23328$ $23328 \div 2 = 11664$ $11664 \div 2 = 5832$ $5832 \div 2 = 2916$ $2916 \div 2 = 1458$ $1458 \div 2 = 729$

Since 729 is not divisible by 2, we now proceed to the next prime number, 3:

$729 \div 3 = 243$ $243 \div 3 = 81$ $81 \div 3 = 27$ $27 \div 3 = 9$ $9 \div 3 = 3$ $3 \div 3 = 1$

So, the prime factorization of 46656 is:

$46656 = 2^6 \times 3^6$

Step 2: Group the factors in triplets

Since we are finding the cube root, we group the factors in sets of three: $46656 = (2^6 \times 3^6) = (2^3)^2 \times (3^3)^2$

Step 3: Take the cube root

Now, we take the cube root by taking one factor from each group: $\sqrt[3]{46656} = 2^2 \times 3^2 = 4 \times 9 = 36$

Final Answer:

The cube root of 46656 is $36$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can we use prime factorization to find the square root of a number?
2. Why do we group factors into triplets when finding the cube root?
3. What is the prime factorization of 729?
4. Can you find the cube root of a number like 5832 using the same method?
5. How is the prime factorization method different from using a calculator for cube roots?

Tip: Prime factorization is helpful for finding cube roots, square roots, and simplifying fractions!