Simplifying a Fraction with Exponents
The expression in the image is a fraction where the numerator is \(10 \times 5^{n+4} + 125 \times 5^{n+2}\) and the denominator is \(3 \times 5^{n+3} + 20 \times 5^{n+1}\). To solve it, you can use the properties of exponents to simplify the expression.
1. Factor out the common \(5^n\) term in both numerator and denominator:
Numerator: \(5^n(10 \times 5^4 + 125 \times 5^2)\)
Denominator: \(5^n(3 \times 5^3 + 20 \times 5^1)\)
2. Now, cancel out the \(5^n\) term from both the numerator and denominator:
Numerator simplifies to: \(10 \times 5^4 + 125 \times 5^2\)
Denominator simplifies to: \(3 \times 5^3 + 20 \times 5^1\)
3. Simplify the coefficients and the remaining powers of 5:
Numerator: \(10 \times 625 + 125 \times 25 = 6250 + 3125 = 9375\)
Denominator: \(3 \times 125 + 20 \times 5 = 375 + 100 = 475\)
4. Simplify the fraction if possible. In this case, 9375 and 475 don't share any common factors other than 1, so the fraction is already in its simplest form:
Final simplified expression: \(\frac{9375}{475}\)
The final answer is a simplified fraction or you could also convert it to a decimal or mixed number if you divide 9375 by 475.
