Question

simplify the expression completely if possible.
\\(\dfrac{2x^2}{x^4 + 9x^3}\\)

Explanation:

Step1: Factor the denominator

First, we factor the denominator \(x^4 + 9x^3\). We can factor out the greatest common factor (GCF) from the terms in the denominator. The GCF of \(x^4\) and \(9x^3\) is \(x^3\). So, we have:
\(x^4 + 9x^3 = x^3(x + 9)\)

Step2: Simplify the fraction

Now, we rewrite the original fraction with the factored denominator:
\(\frac{2x^2}{x^3(x + 9)}\)
We can simplify the \(x\)-terms in the numerator and the denominator. Using the rule of exponents \(\frac{x^a}{x^b}=x^{a - b}\) (where \(a = 2\) and \(b = 3\) here), we get:
\(\frac{2x^2}{x^3(x + 9)}=\frac{2}{x(x + 9)}\) (since \(x^2\div x^3 = x^{2-3}=x^{-1}=\frac{1}{x}\))
We can also expand the denominator if needed, but the simplified form is usually left factored or expanded. Let's expand the denominator: \(x(x + 9)=x^2 + 9x\), so the simplified fraction is \(\frac{2}{x^2 + 9x}\) or \(\frac{2}{x(x + 9)}\).

Answer:

\(\frac{2}{x(x + 9)}\) (or \(\frac{2}{x^2 + 9x}\))