Simplifying Fraction with Common Factors
To simplify the expression given in the image, look for common factors in the numerator and denominator.
The expression is:
\[
\frac{10 \cdot 5^{n+4} + 125 \cdot 5^{n+2}}{3 \cdot 5^{n+3} - 20 \cdot 5^{n+1}}
\]
We can factor out the common powers of 5 from each term and rewrite the expression as follows:
\[
\frac{5^{n+2}(10 \cdot 5^2 + 125)}{5^{n+1}(3 \cdot 5^2 - 20)}
\]
Now simplify inside the parentheses:
\[
\frac{5^{n+2}(10 \cdot 25 + 125)}{5^{n+1}(3 \cdot 25 - 20)}
\]
This simplifies further to:
\[
\frac{5^{n+2}(250 + 125)}{5^{n+1}(75 - 20)}
\]
Combine the numbers inside the parentheses:
\[
\frac{5^{n+2}(375)}{5^{n+1}(55)}
\]
Now that the fractions have a common base of 5, we can cancel out the $5^{n+1}$, leaving $5^1$ in the numerator:
\[
\frac{5 \cdot 375}{55}
\]
Now divide both numerator and denominator by their greatest common divisor, which is 5:
\[
\frac{5 \cdot 75}{11}
\]
This gives us the final simplified result:
\[
\frac{375}{11}
\]
The expression can't be simplified any further, so this is the simplest form of the original expression.
