Simplify (y-8)/(y^2-64)

This mathematical problem is asking for the simplification of a rational expression. The expression provided is a fraction where the numerator is (y-8) and the denominator is a difference of squares (y^2-64). The task involves manipulating the expression to reach its simplest form, which typically means factoring polynomials if possible and then reducing common factors between the numerator and the denominator.

$\frac{y - 8}{y^{2} - 64}$

Answer

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Solution:

Step 1: Factor the denominator.

Step 1.1: Express $64$ as a square of $8$ . $\frac{y - 8}{y^{2} - 8^{2}}$
Step 1.2: Utilize the difference of squares rule, $a^{2} - b^{2} = (a + b) (a - b)$ , where $a = y$ and $b = 8$ . $\frac{y - 8}{(y + 8) (y - 8)}$

Step 2: Eliminate the common term.

Step 2.1: Remove the common term $y - 8$ . $\frac{y - 8}{(y + 8) (y - 8)}$
Step 2.2: Simplify the fraction. $\frac{1}{y + 8}$

Knowledge Notes:

To simplify the given rational expression $(y - 8) / (y^{2} - 64)$ , we can use the following knowledge points:

Difference of Squares: This is a pattern that allows us to factor expressions of the form $a^{2} - b^{2}$ into $(a + b) (a - b)$ . It is applicable here because $y^{2}$ and $64$ are both perfect squares.
Factoring: The process of breaking down an expression into a product of simpler expressions. In this case, we factor the denominator $y^{2} - 64$ into $(y + 8) (y - 8)$ .
Simplifying Rational Expressions: This involves canceling out common factors in the numerator and the denominator of a fraction. Since $y - 8$ is present in both the numerator and the denominator, it can be canceled out.
Final Expression: After canceling the common factors, we are left with a simplified expression, which is easier to understand and work with.

By applying these concepts, we can simplify the original expression to $\frac{1}{y + 8}$ .