Question - Simplifying Algebraic Fraction with Exponents
Solution:
To simplify the expression, we start by simplifying both the numerator and the denominator separately. We can simplify by combining like terms and applying the laws of exponents. Given expression:\[ \frac{10x^{n+4} + 125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \]First, let's simplify the numerator and the denominator by factoring out the common x term:Numerator:\[ 10x^{n+4} + 125x^{n+2} = x^{n+2}(10x^2 + 125) \]Denominator:\[ 3x^{n+3} - 20x^{n+1} = x^{n+1}(3x^2 - 20) \]Now, rewrite the expression using the factored terms:\[ \frac{x^{n+2}(10x^2 + 125)}{x^{n+1}(3x^2 - 20)} \]Next, cancel out the common x term from both the numerator and the denominator: since we have $$x^{n+2}$$ in the numerator and $$x^{n+1}$$ in the denominator, we can divide each by $$x^{n+1}$$ so that:\[ x^{n+2} / x^{n+1} = x \]The simplified expression will be:\[ \frac{x(10x^2 + 125)}{(3x^2 - 20)} \]Now, we can't simplify further without knowing the specific value of $$n$$, as the remaining terms in the parentheses do not have a common x term with which to factor out or cancel. The simplified expression is:\[ x \cdot \frac{10x^2 + 125}{3x^2 - 20} \]
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