Example Question - divide numerator denominator

Simplifying a Fraction with Exponents

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.