problem: What is the value of $b$ if $5^b + 5^b + 5^b + 5^b + 5^b = 625^{(b-1)}$? Express your answer as a common fraction.
solution: We can rewrite $5^b + 5^b + 5^b + 5^b + 5^b$ as $5\cdot5^b=5^{(b+1)}$. Since $625=5^4$, we rewrite $625^{(b-1)}$ as $(5^4)^{(b-1)}=5^{4(b-1)}=5^{(4b-4)}$. We now have $5^{(b+1)}=5^{(4b-4)}$, so it must be that the exponents are equal.  $$b+1=4b-4\qquad\Rightarrow 5=3b\qquad\Rightarrow \frac{5}{3}=b$$ The value of $b$ is $\boxed{\frac{5}{3}}$.