Question - Simplifying Rational Expression with Exponents
Solution:
The expression given in the image needs to be simplified. It's a rational expression involving terms with exponents. To simplify, I'll first rewrite the expression to help clarify the steps:\[ \frac{10x^{5n+4} + 1.25x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \]Now, let's simplify step by step:First, find the common factor in the numerator and denominator to factor out. We can see that $$ x^{5n+1} $$ is the greatest common factor in all terms, which means each term can be expressed as a multiple of $$ x^{5n+1} $$.So, we factor $$ x^{5n+1} $$ out of each term:\[ = \frac{x^{5n+1}(10x^3 + 1.25x)}{x^{5n+1}(3x^2 - 20)} \]Since $$ x^{5n+1} $$ is in both the numerator and the denominator, we can cancel it out:\[ = \frac{10x^3 + 1.25x}{3x^2 - 20} \]The expression is now simplified. However, if you need to further simplify by reducing the coefficients, you can factor constants out or simplify fractions if possible. In this case, the coefficients of the terms in the numerator and denominator don't seem to share any common factors other than 1, so this is the simplest form of the expression:\[ \frac{10x^3 + 1.25x}{3x^2 - 20} \]
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