Question - Simplifying a Rational Expression using Exponent Properties
Solution:
To simplify the given expression, we will apply the properties of exponents. The expression is:\[ \frac{10x^{n+4} + 125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \]First, factor out the common factor of $$x$$ from the numerator and denominator:Numerator: Take $$ x^{n+2} $$ as the common factor.\[ x^{n+2} (10x^2 + 125) \]Denominator: Take $$ x^{n+1} $$ as the common factor.\[ x^{n+1} (3x^2 - 20) \]Now the expression becomes:\[ \frac{x^{n+2} (10x^2 + 125)}{x^{n+1} (3x^2 - 20)} \]Now we will cancel the common $$x$$ terms, remembering that $$x^{n+2} / x^{n+1} = x$$:\[ \frac{x (10x^2 + 125)}{3x^2 - 20} \]After canceling, we are then left with the simplified expression that we cannot further simplify:\[ \frac{x (10x^2 + 125)}{3x^2 - 20} \]This is the simplified version of the original expression, assuming that $$x \neq 0$$ and $$x^2 \neq \frac{20}{3}$$ so that the denominator is not zero.
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