Question

simplify. enter a single, simplified rational expression or integer. \\(\dfrac{\dfrac{6}{x^{19}} - \dfrac{1}{5}}{\dfrac{x - 6}{10}}\\)

Explanation:

Step1: Simplify the numerator

First, simplify the numerator \(\frac{6}{x^{19}}-\frac{1}{5}\). Find a common denominator, which is \(5x^{19}\). Then we have:

$$ \frac{6\times5 - x^{19}}{5x^{19}}=\frac{30 - x^{19}}{5x^{19}} $$

Step2: Rewrite the complex fraction

The original expression is a complex fraction \(\frac{\frac{30 - x^{19}}{5x^{19}}}{\frac{x - 6}{10}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so we get:

$$ \frac{30 - x^{19}}{5x^{19}}\times\frac{10}{x - 6} $$

Step3: Simplify the expression

Simplify the constants: \(\frac{10}{5} = 2\). So the expression becomes:

$$ \frac{2(30 - x^{19})}{x^{19}(x - 6)}=\frac{60 - 2x^{19}}{x^{19}(x - 6)} $$

We can also factor out a \(- 2\) from the numerator:

$$ \frac{-2(x^{19}-30)}{x^{19}(x - 6)}=\frac{2(30 - x^{19})}{x^{19}(x - 6)} $$

Answer:

\(\frac{2(30 - x^{19})}{x^{19}(x - 6)}\) (or \(\frac{60 - 2x^{19}}{x^{20}-6x^{19}}\) after expanding the denominator)