Sure, let's break down the expression [tex]\((\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y})\)[/tex] and simplify it step by step.
This expression is a classic example that can be simplified using the difference of squares formula. The difference of squares formula states:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
In the given expression [tex]\(\sqrt{x} - \sqrt{y}\)[/tex] and [tex]\(\sqrt{x} + \sqrt{y}\)[/tex], we can identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:
- [tex]\(a = \sqrt{x}\)[/tex]
- [tex]\(b = \sqrt{y}\)[/tex]
According to the difference of squares formula:
[tex]\[
(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y}) = (\sqrt{x})^2 - (\sqrt{y})^2
\][/tex]
Next, we calculate the squares of the square roots:
- [tex]\((\sqrt{x})^2 = x\)[/tex]
- [tex]\((\sqrt{y})^2 = y\)[/tex]
Putting these back into the expression, we get:
[tex]\[
(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y}) = x - y
\][/tex]
Thus, the simplified expression is:
[tex]\[
x - y
\][/tex]
Therefore, [tex]\(\boxed{x - y}\)[/tex] is the simplified result.
