To determine if [tex]\(7^2 \cdot 11^5\)[/tex] is a factor of [tex]\(7^3 \cdot 11^5\)[/tex], let's analyze the exponents on each of the bases (7 and 11) in the given expressions.
1. Analyzing the factor of 7:
- In the expression [tex]\(7^2 \cdot 11^5\)[/tex], the exponent of 7 is 2.
- In the expression [tex]\(7^3 \cdot 11^5\)[/tex], the exponent of 7 is 3.
- For [tex]\(7^2\)[/tex] to be a factor of [tex]\(7^3\)[/tex], the exponent in [tex]\(7^2\)[/tex] (which is 2) must be less than or equal to the exponent in [tex]\(7^3\)[/tex] (which is 3). Clearly, 2 ≤ 3, so this condition is satisfied.
2. Analyzing the factor of 11:
- In the expression [tex]\(7^2 \cdot 11^5\)[/tex], the exponent of 11 is 5.
- In the expression [tex]\(7^3 \cdot 11^5\)[/tex], the exponent of 11 is also 5.
- For [tex]\(11^5\)[/tex] to be a factor of [tex]\(11^5\)[/tex], the exponents must be equal, which they are (both exponents are 5). So this condition is also satisfied.
Since both conditions are satisfied, [tex]\(7^2 \cdot 11^5\)[/tex] is indeed a factor of [tex]\(7^3 \cdot 11^5\)[/tex].
To confirm, you can see that:
[tex]\[ 7^3 \cdot 11^5 = 7 \cdot 7^2 \cdot 11^5 \][/tex]
which confirms that [tex]\(7^2 \cdot 11^5\)[/tex] is included within [tex]\(7^3 \cdot 11^5\)[/tex], verifying it is a factor.
Therefore, the correct choice is:
B. Yes, because [tex]\(7^3 \cdot 11^5=7 \cdot 7^2 \cdot 11^5\)[/tex].
