To determine which expression is equivalent to [tex]\(\frac{7^{13}}{7^7}\)[/tex], we need to simplify the given expression using the laws of exponents.
Given:
[tex]\[
\frac{7^{13}}{7^7}
\][/tex]
Recall the property of exponents that states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Here, [tex]\(a\)[/tex] is the base (7), [tex]\(m\)[/tex] is the exponent in the numerator (13), and [tex]\(n\)[/tex] is the exponent in the denominator (7).
Applying the property:
[tex]\[
\frac{7^{13}}{7^7} = 7^{13-7}
\][/tex]
Subtract the exponents:
[tex]\[
7^{13-7} = 7^6
\][/tex]
So, the expression [tex]\(\frac{7^{13}}{7^7}\)[/tex] simplifies to [tex]\(7^6\)[/tex].
Therefore, the equivalent expression is:
[tex]\[
\boxed{7^6}
\][/tex]
Among the given options:
A. [tex]\(7^{91}\)[/tex]
B. [tex]\(7^6\)[/tex]
C. [tex]\(7^5\)[/tex]
D. [tex]\(7^{20}\)[/tex]
The correct answer is:
[tex]\[
\boxed{7^6}
\][/tex]
