Question - Simplifying a Fraction with Exponents
mathsimplified expressionsimplify fraction exponentsproperties of exponentscommon factors in fractions
Solution:
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.
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