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

its wrong

Answer 2

To express the number 10032 as a product of identical factors, we first need to find its prime factorization.

1. Divide 10032 by small prime numbers:
- 10032 is even, so divide by 2:
\[
10032 \div 2 = 5016
\]
- Continuing with 2:
\[
5016 \div 2 = 2508
\]
- Continuing with 2:
\[
2508 \div 2 = 1254
\]
- Continuing with 2:
\[
1254 \div 2 = 627
\]
- Now, 627 is not divisible by 2. Check 3:
\[
627 \div 3 = 209
\]
- 209 is not divisible by 3. Check 11 (the next smallest prime):
\[
209 \div 11 = 19
\]
- 19 is a prime number.

So the complete prime factorization of 10032 is:
\[
10032 = 2^4 \times 3^1 \times 11^1 \times 19^1
\]

To express this as a product of identical factors, we need to determine the largest identical factor. The factor with the highest exponent here is \( 2 \), so we can rewrite the expression as follows.

Since \( 2^4 = 16 \), we can find pairs:
\[
10032 = (2^2 \times 3 \times 11 \times 19) \times (2^2) = (4 \times 3 \times 11 \times 19) \times 4
\]

Calculating \( 4 \times 3 \times 11 \times 19 \):
- First, \( 3 \times 11 = 33 \)
- Then, \( 33 \times 19 = 627 \)
- Finally, \( 4 \times 627 = 2508 \)

Thus:
\[
10032 = 2508 \times 4
\]

In simplified form, the product of identical factors for 10032 could be expressed as:
\[
(4 \times 3 \times 11 \times 19) \times 4
\]

This yields:
\[
10032 = (4 \times 627) \rightarrow (4^4) \text{ (since 4 is the simplest identical factor)}
\]

Thus, \( 10032 \) can ultimately be simplified as:
\[
= (2^2) \cdot (2^2 \cdot 3 \cdot 11 \cdot 19)^4
\]

Putting this together yields:

\[
10032 = (2^2) \cdot (3 \times 11 \times 19)^4
\]

Thus the product of identical factors is:
\[
(4 \cdot 627) \times 4 \text{ or simply \( 10032 \) expressed in its simplest factor form: }
\]
\[
2^4 \cdot 3^1 \cdot 11^1 \cdot 19^1.
\]