Rewrite the expression as a product of identical factors to find the product for 10032

Question

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(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):

   • $10032÷2=5016$
   • $5016÷2=2508$
   • $2508÷2=1254$
   • $1254÷2=627$

2. Now divide by 3 (the next smallest prime number):

   • $627÷3=209$

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: $$10032=32\times 313;(breakthehighercompositesleadingtofullresolution)$$

Ultimately then: $$\boxed{{\displaystyle 10032=16\times 627}}$$ and ensuring square factor focus: $$=2^4\times 3\times 209leadingthroughidentitiestoproperidentification.$$

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