To show that the expression [tex]\((3 - \sqrt{8})(5 + \sqrt{18})\)[/tex] can be written in the form [tex]\(a + b\sqrt{2}\)[/tex], let's simplify the given expression step by step:
1. Express the square roots with common factors:
- [tex]\(\sqrt{8}\)[/tex] can be written as [tex]\(\sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}\)[/tex].
- [tex]\(\sqrt{18}\)[/tex] can be written as [tex]\(\sqrt{9 \cdot 2} = \sqrt{9}\sqrt{2} = 3\sqrt{2}\)[/tex].
So, rewrite the original expression with these simplified radicals:
[tex]\[(3 - 2\sqrt{2})(5 + 3\sqrt{2})\][/tex]
2. Use the distributive property (FOIL method) to expand the expression:
[tex]\[
(3 - 2\sqrt{2})(5 + 3\sqrt{2}) = 3 \cdot 5 + 3 \cdot 3\sqrt{2} - 2\sqrt{2} \cdot 5 - 2\sqrt{2} \cdot 3\sqrt{2}
\][/tex]
3. Calculate each term in the expansion:
- First term: [tex]\(3 \cdot 5 = 15\)[/tex]
- Second term: [tex]\(3 \cdot 3\sqrt{2} = 9\sqrt{2}\)[/tex]
- Third term: [tex]\(-2\sqrt{2} \cdot 5 = -10\sqrt{2}\)[/tex]
- Fourth term: [tex]\(-2\sqrt{2} \cdot 3\sqrt{2} = -2 \cdot 3 \cdot (\sqrt{2})^2 = -6 \cdot 2 = -12\)[/tex]
Combine these results:
[tex]\[
15 + 9\sqrt{2} - 10\sqrt{2} - 12
\][/tex]
4. Simplify the expression by combining like terms:
- Combine the constant terms: [tex]\(15 - 12 = 3\)[/tex]
- Combine the [tex]\(\sqrt{2}\)[/tex] terms: [tex]\(9\sqrt{2} - 10\sqrt{2} = -\sqrt{2}\)[/tex]
So, the expanded and simplified expression is:
[tex]\[
3 - \sqrt{2}
\][/tex]
Therefore, the expression [tex]\((3 - \sqrt{8})(5 + \sqrt{18})\)[/tex] can indeed be written in the form [tex]\(a + b\sqrt{2}\)[/tex], where [tex]\(a = 3\)[/tex] and [tex]\(b = -1\)[/tex].
