Question - Simplifying Fraction with Common Factors
Solution:
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.
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