The problem asks us to simplify the expression $\frac{5^7}{5^8}$.

1. Problem Description

The problem asks us to simplify the expression $\frac{5^7}{5^8}$.

2. Solution Steps

We can use the quotient rule for exponents to simplify the expression. The quotient rule states that when dividing exponents with the same base, we subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$

In this case, $a = 5$, $m = 7$, and $n = 8$. Applying the rule:

$\frac{5^7}{5^8} = 5^{7-8} = 5^{-1}$

Since $a^{-n} = \frac{1}{a^n}$, we can rewrite $5^{-1}$ as $\frac{1}{5^1}$.

$5^{-1} = \frac{1}{5^1} = \frac{1}{5}$

3. Final Answer

$\frac{1}{5}$