Which of the following numbers are rational?
`(4-sqrt(2))(2+1/sqrt(2))`

Answer

To determine if the expression \((4 - \sqrt{2})(2 + \frac{1}{\sqrt{2}})\) is rational, we will simplify it step by step.

### Step 1: Expand the Expression
We start by expanding the expression using the distributive property (also known as the FOIL method for binomials):

\[ (4 - \sqrt{2})(2 + \frac{1}{\sqrt{2}}) = 4 \cdot 2 + 4 \cdot \frac{1}{\sqrt{2}} - \sqrt{2} \cdot 2 - \sqrt{2} \cdot \frac{1}{\sqrt{2}} \]

### Step 2: Calculate Each Term
Now we calculate each term:

1. \(4 \cdot 2 = 8\)
2. \(4 \cdot \frac{1}{\sqrt{2}} = \frac{4}{\sqrt{2}}\)
3. \(-\sqrt{2} \cdot 2 = -2\sqrt{2}\)
4. \(-\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -1\) (since \(\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1\))

Putting it all together, we have:

\[ 8 + \frac{4}{\sqrt{2}} - 2\sqrt{2} - 1 \]

### Step 3: Combine Like Terms
Now we combine the constant terms and the terms involving \(\sqrt{2}\):

1. Combine the constant terms: \(8 - 1 = 7\)
2. The term \(\frac{4}{\sqrt{2}}\) can be simplified. We can multiply the numerator and denominator by \(\sqrt{2}\):

\[ \frac{4}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \]

Now we have:

\[ 7 + 2\sqrt{2} - 2\sqrt{2} \]

### Step 4: Simplify Further
Notice that \(2\sqrt{2} - 2\sqrt{2} = 0\). So we are left with:

\[ 7 + 0 = 7 \]

### Conclusion
The final result is \(7\), which is a rational number.

### Final Answer
Thus, the expression \((4 - \sqrt{2})(2 + \frac{1}{\sqrt{2}})\) simplifies to \(7\), which is a rational number.

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