Question - Simplifying a Fraction with Exponents
Solution:
The expression given is:\[ \frac{10 \cdot 5^{n+4} + 125 \cdot 5^{n+2}}{3 \cdot 5^{n+3} - 20 \cdot 5^{n+1}} \]To simplify, we'll factor out the common base of $$ 5^n $$ from both the numerator and the denominator:\[ = \frac{5^n (10 \cdot 5^4 + 125 \cdot 5^2)}{5^n (3 \cdot 5^3 - 20 \cdot 5)} \]\[ = \frac{5^n (10 \cdot 625 + 125 \cdot 25)}{5^n (3 \cdot 125 - 20 \cdot 5)} \]\[ = \frac{5^n (6250 + 3125)}{5^n (375 - 100)} \]\[ = \frac{5^n \cdot 9375}{5^n \cdot 275} \]Since $$ 5^n $$ is a common factor in both the numerator and the denominator, we can cancel it out:\[ = \frac{9375}{275} \]Now we'll divide both the numerator and the denominator by 25 to simplify further:\[ = \frac{375}{11} \]The simplified form of the expression is $$ \frac{375}{11} $$, which cannot be simplified any further to an integer or a reduced fraction.
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