Question - Simplified Expression with Factoring
Solution:
To simplify the given expression, we will factor common terms and then reduce wherever possible. The expression is:(10 * 5^(n+4) + 125 * 5^(n+2)) / (3 * 5^(n+3) - 20 * 5^(n+1))Let's factor out the common 5 raised to the smallest power in both the numerator and the denominator.For the numerator, the smallest power of 5 is n+2. We can then express each term with 5^(n+2) factored out:5^(n+2) * (10 * 5^2 + 125)For the denominator, the smallest power of 5 is n+1. Factor out 5^(n+1):5^(n+1) * (3 * 5^2 - 20)Now, we can substitute the powers of 5 with its actual value:Numerator becomes:5^(n+2) * (10 * 25 + 125)5^(n+2) * (250 + 125)5^(n+2) * 375Denominator becomes:5^(n+1) * (3 * 25 - 20)5^(n+1) * (75 - 20)5^(n+1) * 55Now, put the numerator and denominator back together:(5^(n+2) * 375) / (5^(n+1) * 55)Simplify by cancelling out the common factor of 5:5^1 in the numerator remains after cancelling out with 5^(n+1) in the denominator and then also simplify the numbers:5 * 375 / 55Now, simplifying the numerical part, 375/55 can be reduced:5 * 375 / 55 = 5 * 75 / 11The final simplified expression is:(5 * 75) / 11 or 375 / 11
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