Question - Simplifying Algebraic Fraction with Common Powers of 5
Solution:
The expression given is:$$ \frac{10 \times 5^{n+4} + 125 \times 5^{n+2}}{3 \times 5^{n+3} - 20 \times 5^{n+1}} $$We can simplify this expression by factoring common terms and any powers of 5 that appear in all terms, with the goal of simplifying the fraction by cancellation where possible.Let's factor out the greatest power of 5 from the numerator and denominator.The numerator has $$ 5^{n+4} $$ and $$ 5^{n+2} $$, so we can factor out $$ 5^{n+2} $$:$$ 5^{n+2} \times (10 \times 5^2 + 125) $$$$ = 5^{n+2} \times (10 \times 25 + 125) $$$$ = 5^{n+2} \times (250 + 125) $$$$ = 5^{n+2} \times 375 $$The denominator has $$ 5^{n+3} $$ and $$ 5^{n+1} $$, so we can factor out $$ 5^{n+1} $$:$$ 5^{n+1} \times (3 \times 5^2 - 20) $$$$ = 5^{n+1} \times (3 \times 25 - 20) $$$$ = 5^{n+1} \times (75 - 20) $$$$ = 5^{n+1} \times 55 $$After factoring out the common powers of 5, the original expression now looks like this:$$ \frac{5^{n+2} \times 375}{5^{n+1} \times 55} $$We can cancel out a $$ 5^{n+1} $$ from the numerator and denominator, leaving us with:$$ \frac{5 \times 375}{55} $$Divide 375 by 55:$$ \frac{5 \times 375}{55} = \frac{5 \times 15}{11} $$So, the simplified form of the original expression is:$$ \frac{75}{11} $$This fraction cannot be simplified further as 75 and 11 are relatively prime (their greatest common divisor is 1).
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