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
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 $y - 8$. $\frac{\cancel{y - 8}}{(y + 8)\cancel{(y - 8)}}$
- 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 $\frac{1}{y + 8}$.
