Simplifying a Rational Expression using Exponent Properties
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.
