To solve for the value of [tex]\( b \)[/tex] in the given expression [tex]\((3 - 4 \sqrt{2})(1 + 3 \sqrt{2}) = a + b \sqrt{2}\)[/tex], we need to expand the left-hand side and then compare coefficients of like terms on both sides of the equation. Here are the detailed steps:
### Step 1: Expand the Expression
We start by expanding [tex]\((3 - 4 \sqrt{2})(1 + 3 \sqrt{2})\)[/tex] using the distributive property (also known as the FOIL method for binomials):
[tex]\[ (3 - 4\sqrt{2})(1 + 3\sqrt{2}) \][/tex]
First, distribute each term in the first binomial to each term in the second binomial:
1. Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[ 3 \times 1 = 3 \][/tex]
2. Multiply [tex]\(3\)[/tex] by [tex]\(3\sqrt{2}\)[/tex]:
[tex]\[ 3 \times 3\sqrt{2} = 9\sqrt{2} \][/tex]
3. Multiply [tex]\(-4\sqrt{2}\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[ -4\sqrt{2} \times 1 = -4\sqrt{2} \][/tex]
4. Multiply [tex]\(-4\sqrt{2}\)[/tex] by [tex]\(3\sqrt{2}\)[/tex]:
[tex]\[ -4\sqrt{2} \times 3\sqrt{2} = -4 \times 3 \times (\sqrt{2})^2 = -12 \times 2 = -24 \][/tex]
Now combine all these results together:
[tex]\[ 3 + 9\sqrt{2} - 4\sqrt{2} - 24 \][/tex]
### Step 2: Simplify the Expression
Combine the like terms (those that are rational numbers and those that involve [tex]\(\sqrt{2}\)[/tex]):
[tex]\[ (3 - 24) + (9\sqrt{2} - 4\sqrt{2}) \][/tex]
[tex]\[ -21 + 5\sqrt{2} \][/tex]
### Step 3: Compare Coefficients
We have the simplified expression:
[tex]\[ -21 + 5\sqrt{2} \][/tex]
This must be equal to [tex]\(a + b\sqrt{2}\)[/tex]. By comparing coefficients, the term involving [tex]\(\sqrt{2}\)[/tex] gives us:
[tex]\[ b = 5 \][/tex]
Therefore, the value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
