To solve the division of the given expressions:
[tex]\[
\frac{y^2-4}{y} \div \frac{y+2}{y-2}
\][/tex]
we follow these steps:
### Step 1: Rewrite the Division as Multiplication
Remember that dividing by a fraction is equivalent to multiplying by its reciprocal. So, rewrite the expression as:
[tex]\[
\frac{y^2-4}{y} \times \frac{y-2}{y+2}
\][/tex]
### Step 2: Factorize the Expressions
Factorize any quadratic expressions where possible. Notice that [tex]\( y^2 - 4 \)[/tex] is a difference of squares and can be factored as:
[tex]\[
y^2 - 4 = (y + 2)(y - 2)
\][/tex]
Using this factorization, the expression becomes:
[tex]\[
\frac{(y+2)(y-2)}{y} \times \frac{y-2}{y+2}
\][/tex]
### Step 3: Simplify the Multiplication
Combine the numeric and algebraic fractions:
[tex]\[
\frac{(y+2)(y-2)}{y} \times \frac{y-2}{y+2}
\][/tex]
Notice that the term [tex]\( (y+2) \)[/tex] in the numerator of the first fraction and the denominator of the second fraction can cancel each other out. Similarly, the [tex]\( (y-2) \)[/tex] terms can also be cancelled out, but carefully handle terms that might simplify out entirely. The expression simplifies to:
[tex]\[
\frac{(y-2)}{y} \times (y-2)
\][/tex]
Which becomes:
[tex]\[
\frac{(y-2)(y-2)}{y}
\][/tex]
### Step 4: Distribute and Simplify
Expand the numerator:
[tex]\[
\frac{(y-2)^2}{y}
\][/tex]
This can be written as:
[tex]\[
\frac{y^2 - 4y + 4}{y}
\][/tex]
Then, separate each term in the numerator:
[tex]\[
\frac{y^2}{y} - \frac{4y}{y} + \frac{4}{y}
\][/tex]
Which simplifies to:
[tex]\[
y - 4 + \frac{4}{y}
\][/tex]
### Final Answer
Thus, dividing the given expressions results in:
[tex]\[
y - 4 + \frac{4}{y}
\][/tex]
