To solve the problem of finding the prime factorization of [tex]\(670^2\)[/tex], let's break it down step by step.
1. Square the Given Number:
We start with the number 670. Squaring this number gives:
[tex]\[
670^2 = 448900
\][/tex]
2. Prime Factorization of 670:
Next, we find the prime factorization of the number 670.
[tex]\[
670 = 2^1 \cdot 5^1 \cdot 67^1
\][/tex]
Here, 670 is factored into primes as 2, 5, and 67.
3. Square Each Prime Factor:
When you square 670, all of its prime factors must also be squared. Therefore:
[tex]\[
(2 \cdot 5 \cdot 67)^2
\][/tex]
This leads to squaring each individual prime factor, resulting in:
[tex]\[
(2^1)^2 \cdot (5^1)^2 \cdot (67^1)^2 = 2^2 \cdot 5^2 \cdot 67^2
\][/tex]
4. Compile the Prime Factorization:
After performing the above calculations, the complete prime factorization of [tex]\(670^2\)[/tex] is:
[tex]\[
2^2 \cdot 5^2 \cdot 67^2
\][/tex]
### Comparison with Given Choices:
Given the choices:
1. [tex]\(2^2 \cdot 5^2 \cdot 67^2\)[/tex]
2. [tex]\(5^2 \cdot 67^2\)[/tex]
3. [tex]\(10^2 \cdot 67^2\)[/tex]
4. [tex]\(2^4 \cdot 67^2\)[/tex]
The correct prime factorization of [tex]\(670^2\)[/tex] as determined is:
[tex]\[
2^2 \cdot 5^2 \cdot 67^2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2^2 \cdot 5^2 \cdot 67^2}
\][/tex]
