To solve [tex]\((\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})\)[/tex], we utilize the difference of squares formula, which states that:
[tex]$(a - b)(a + b) = a^2 - b^2$[/tex]
In this case, we can identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:
- [tex]\(a = \sqrt{5}\)[/tex]
- [tex]\(b = \sqrt{2}\)[/tex]
Applying the difference of squares formula, we get:
[tex]$(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2$[/tex]
Next, we calculate each term separately:
1. Calculate [tex]\((\sqrt{5})^2\)[/tex]:
[tex]\[
(\sqrt{5})^2 = 5
\][/tex]
2. Calculate [tex]\((\sqrt{2})^2\)[/tex]:
[tex]\[
(\sqrt{2})^2 = 2
\][/tex]
Now, subtract the second result from the first:
[tex]\[
5 - 2 = 3
\][/tex]
Therefore, the value of [tex]\((\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})\)[/tex] is:
[tex]\[
3.0000000000000004
\][/tex]
This slight deviation from 3 is due to the precision of numerical calculations, but for practical purposes, it is effectively 3.
