Question - Algebraic Expression Simplification with Exponent Rules
Solution:
This is an algebraic expression simplification problem involving exponent rules. We'll simplify each term by factoring out the common powers of 5 from the numerator and the denominator.Given expression:$$ \frac{10 \times 5^{n+4} + 125 \times 5^{n+2}}{3 \times 5^{n+3} - 20 \times 5^{n+1}} $$Step 1: Factor out common powers of 5 from each term.Numerator:- Factor out $$ 5^{n+2} $$ from each term.$$ 5^{n+2} (10 \times 5^2 + 125) = 5^{n+2} (10 \times 25 + 125) = 5^{n+2} (250 + 125) = 5^{n+2} \times 375 $$Denominator:- Factor out $$ 5^{n+1} $$ from each term.$$ 5^{n+1} (3 \times 5^2 - 20) = 5^{n+1} (3 \times 25 - 20) = 5^{n+1} (75 - 20) = 5^{n+1} \times 55 $$Step 2: Simplify the expression with the factored out powers of 5.The simplified expression is:$$ \frac{5^{n+2} \times 375}{5^{n+1} \times 55} $$Step 3: Cancel out the common power of 5 from the numerator and denominator.$$ 5^{n+2} $$ in the numerator and $$ 5^{n+1} $$ in the denominator have a common base, which means we can subtract the exponents.$$ 5^{n+2 - (n+1)} = 5^1 = 5 $$So, the simplification yields:$$ \frac{5 \times 375}{55} $$Step 4: Simplify the coefficients (375/55).Divide 375 by 55 to get:$$ \frac{375}{55} = \frac{75}{11} $$So the final simplified expression is:$$ \frac{5 \times 75}{11} $$Step 5: Multiply the 5 by 75.$$ 5 \times 75 = 375 $$Therefore, the final simplified form of the expression is:$$ \frac{375}{11} $$ This fraction cannot be simplified further, as 375 and 11 do not have any common factors other than 1.
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