Sure, let's multiply the two binomials [tex]\((\sqrt{10} + 2\sqrt{8})(\sqrt{10} - 2\sqrt{8})\)[/tex] step-by-step and find the final result.
We can use the difference of squares formula to simplify this expression. The difference of squares formula is:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
In our case:
- [tex]\(a = \sqrt{10}\)[/tex]
- [tex]\(b = 2\sqrt{8}\)[/tex]
Now, substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
[tex]\[
(\sqrt{10} + 2\sqrt{8})(\sqrt{10} - 2\sqrt{8}) = (\sqrt{10})^2 - (2\sqrt{8})^2
\][/tex]
Next, we calculate each term separately:
1. [tex]\((\sqrt{10})^2\)[/tex]:
[tex]\[
(\sqrt{10})^2 = 10
\][/tex]
2. [tex]\((2\sqrt{8})^2\)[/tex]:
[tex]\[
(2\sqrt{8})^2 = 2^2 \cdot (\sqrt{8})^2 = 4 \cdot 8 = 32
\][/tex]
Subtracting these terms:
[tex]\[
10 - 32 = -22
\][/tex]
So, the final result of [tex]\((\sqrt{10} + 2\sqrt{8})(\sqrt{10} - 2\sqrt{8})\)[/tex] is:
[tex]\[
\boxed{-22}
\][/tex]
