Simplify (x-13)/(x^2-169)

Solution:

Step:1

Identify and simplify the denominator.

Step:1.1

Express $169$ as a square of $13$: $\left(13\right)^{2}$. The expression becomes $\frac{x - 13}{x^{2} - \left(13\right)^{2}}$.

Step:1.2

Apply the difference of squares identity, $a^{2} - b^{2} = \left(a + b\right) \left(a - b\right)$, with $a = x$ and $b = 13$. This yields $\frac{x - 13}{\left(x + 13\right) \left(x - 13\right)}$.

Step:2

Eliminate the common term in the numerator and the denominator.

Step:2.1

Remove the common term $x - 13$: $\frac{\cancel{x - 13}}{\left(x + 13\right) \cancel{x - 13}}$.

Step:2.2

Simplify the fraction to its reduced form: $\frac{1}{x + 13}$.

Knowledge Notes:

To solve the given problem, we need to understand several mathematical concepts:

1. Difference of Squares: This is a pattern used to factor expressions of the form $a^{2} - b^{2}$ into $\left(a + b\right) \left(a - b\right)$. It is applicable when both terms are perfect squares.

2. Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this case, the denominator $x^{2} - 169$ is factored using the difference of squares.

3. Simplifying Rational Expressions: This involves reducing a fraction to its simplest form by eliminating common factors in the numerator and the denominator.

4. Perfect Squares: These are numbers or expressions that are the square of an integer or a simpler expression, such as $13^{2} = 169$.

5. Cancellation: In a fraction, if the same term appears in both the numerator and the denominator, it can be cancelled out, simplifying the fraction.

By applying these concepts, we can simplify the given rational expression from $\frac{x - 13}{x^{2} - 169}$ to $\frac{1}{x + 13}$.