Question - Simplifying a Rational Expression with Exponents
mathcombining like termssimplify rational expressionsexponents with expressionsproperties of exponents
Solution:
To simplify the given expression, we will use properties of exponents and combine like terms. Here's the expression, with each term separated for clarity:\[ \frac{10x^{5n+4} + 4.125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \]First, let's look at the exponents in the numerator and the denominator to see if we can simplify by factoring out common factors. We want to identify the term with the lowest exponent for $$ x $$ that is common to all terms.In this case, the term with the lowest exponent for $$ x $$ is $$ x^{5n+1} $$, so let's factor that out from each term:Numerator:\[ 10x^{5n+4} = 10x^{5n+1} \cdot x^3 \]\[ 4.125x^{5n+2} = 4.125x^{5n+1} \cdot x \]Denominator:\[ 3x^{5n+3} = 3x^{5n+1} \cdot x^2 \]\[ 20x^{5n+1} \]We can now rewrite the expression:\[ \frac{x^{5n+1} (10x^3 + 4.125x)}{x^{5n+1} (3x^2 - 20)} \]We simplify by canceling the common factor $$ x^{5n+1} $$ from both the numerator and the denominator:\[ \frac{10x^3 + 4.125x}{3x^2 - 20} \]The expression is now simplified to the extent that we can without further information about $$ x $$. We cannot simplify it further algebraically, as there are no common factors left to cancel between the numerator and the denominator. The result is a rational expression in terms of $$ x $$.
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