Sure! Let's break down the process of expressing [tex]\(105^2\)[/tex] as a product of its prime factors step-by-step.
1. Identify the prime factors of 105:
105 can be factorized into prime numbers. The prime factorization of 105 is:
[tex]\[
105 = 3 \times 5 \times 7
\][/tex]
2. Square 105:
To find [tex]\(105^2\)[/tex], we simply multiply 105 by itself:
[tex]\[
105^2 = 105 \times 105 = 11025
\][/tex]
3. Identify the prime factors of 105 squared:
Since [tex]\(105 = 3 \times 5 \times 7\)[/tex], we can square each prime factor:
[tex]\[
105^2 = (3 \times 5 \times 7)^2
\][/tex]
4. Distribute the square to each prime factor:
Distributing the square to each of the factors, we get:
[tex]\[
105^2 = 3^2 \times 5^2 \times 7^2
\][/tex]
5. Write [tex]\(105^2\)[/tex] as a product of its prime factors:
Finally, combining these, we get:
[tex]\[
105^2 = 3^2 \times 5^2 \times 7^2
\][/tex]
Therefore, [tex]\(105^2\)[/tex] expressed as a product of its prime factors is [tex]\(3^2 \times 5^2 \times 7^2\)[/tex].
