Question

Rewrite the expression as a product of identical factors to find the product for 10032
. Submit your answer in simplified form. /././



(1 point)

Answer 1

To rewrite the expression \(10032\) as a product of identical factors, we need to factor it completely. Let's start with the prime factorization of \(10032\).

1. **Divide by 2** (the smallest prime number):
- \(10032 \div 2 = 5016\)
- \(5016 \div 2 = 2508\)
- \(2508 \div 2 = 1254\)
- \(1254 \div 2 = 627\)

2. **Now divide by 3** (the next smallest prime number):
- \(627 \div 3 = 209\)

3. **209 is not divisible by 2 or 3, so check for 11 and find it is prime**.

Now we can summarize the factorization steps:
\[
10032 = 2^4 \times 3^1 \times 209^1
\]

Next, let's find the largest identical factor we can express. The largest factor breakdown with identical components is to combine the repeated factors:

- \(10032 = 2^4 \times 3^1 \times 209^1 = (2^2)^4 \times 3^1 \times 209^1 = 4^2 \times 3 \times 209\)
- This suggests trying to express in simpler multiples.

Let’s express \(10032\) as a product of smaller factors, we can calculate:

1. Taking pairs from our factorization, we simplify aiming towards square roots to find the prime factor counts:
\[
= 16 \times 3 \times 209
\]

Through correction pairs:
\[
= 48 \times 209
\]

Continuing to organize and simplify:
\[
= 10032 = (12 \times 4) \times (3 \times 209) \text{ providing structured factors}
\]

Thus, the expression can be rearranged and confirmed:
\[
10032 = (48)^2 \text{ or} \; (4 \times 3 \times 209) = 10032
\]

Final confirmation is performed for simplification:
\[
10032 = 32 \times 313 \; (break the higher composites leading to full resolution)
\]

Ultimately then:
\[
\boxed{10032 = 16 \times 627}
\]
and ensuring square factor focus:
\[
= 2^4 \times 3 \times 209 leading through identities to proper identification.
\]

This product can be confirmed as necessary for clarity in final resolution paths.