Question

perform the indicated operation and simplify. \\(\frac{7}{60x} + \frac{3}{60x} = \\) \\(\frac{x^2}{x - 8} - \frac{13x - 40}{x - 8} = \\)

Explanation:

Step1: Add numerators (common denominator)

Since the fractions $\frac{7}{60x}$ and $\frac{3}{60x}$ have the same denominator, add their numerators directly.
$\frac{7 + 3}{60x}$

Step2: Simplify the first expression

Calculate the sum in the numerator.
$\frac{10}{60x} = \frac{1}{6x}$

Step3: Subtract numerators (common denominator)

For $\frac{x^2}{x-8} - \frac{13x-40}{x-8}$, use the common denominator to subtract numerators.
$\frac{x^2 - (13x - 40)}{x-8}$

Step4: Simplify the numerator

Expand and combine like terms in the numerator.
$\frac{x^2 - 13x + 40}{x-8}$

Step5: Factor the numerator

Factor the quadratic expression in the numerator.
$\frac{(x-5)(x-8)}{x-8}$

Step6: Cancel common factors

Cancel the $(x-8)$ term from numerator and denominator (where $x
eq 8$).
$x - 5$

Answer:

$\frac{1}{6x}$
$x - 5$