Question

fully simplify.
\\(\dfrac{\dfrac{x}{5} + 1}{\dfrac{5}{x^2} - \dfrac{1}{5}}\\)

Explanation:

Step1: Simplify numerator and denominator separately

First, simplify the numerator \(\frac{x}{5}+1\). We can rewrite \(1\) as \(\frac{5}{5}\), so \(\frac{x}{5}+\frac{5}{5}=\frac{x + 5}{5}\).

Next, simplify the denominator \(\frac{5}{x^{2}}-\frac{1}{5}\). To subtract these fractions, we find a common denominator, which is \(5x^{2}\). So we have \(\frac{5\times5}{5x^{2}}-\frac{1\times x^{2}}{5x^{2}}=\frac{25 - x^{2}}{5x^{2}}\). Notice that \(25 - x^{2}\) is a difference of squares, so it can be factored as \((5 - x)(5 + x)\) or \(-(x - 5)(x + 5)\).

Step2: Rewrite the complex fraction as a division of two fractions

The original expression \(\frac{\frac{x}{5}+1}{\frac{5}{x^{2}}-\frac{1}{5}}\) is now \(\frac{\frac{x + 5}{5}}{\frac{25 - x^{2}}{5x^{2}}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes \(\frac{x + 5}{5}\times\frac{5x^{2}}{25 - x^{2}}\).

Step3: Cancel common factors and simplify

We can cancel the \(5\) in the numerator and denominator. Also, since \(25 - x^{2}=(5 - x)(5 + x)\), we can rewrite the expression as \(\frac{x + 5}{1}\times\frac{x^{2}}{(5 - x)(5 + x)}\). The \((x + 5)\) terms cancel out (assuming \(x
eq - 5\) to avoid division by zero), leaving us with \(\frac{x^{2}}{5 - x}\) or we can factor out a negative sign from the denominator to get \(-\frac{x^{2}}{x - 5}\).

Answer:

\(\boldsymbol{\frac{x^{2}}{5 - x}}\) (or equivalently \(\boldsymbol{-\frac{x^{2}}{x - 5}}\))