Algebraic Expression Simplification with Exponent Rules
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.
