If `x= 3 + 2 sqrt2`, find
`(1)/(x)`

Answer

To find the value of \( \frac{1}{x} \) where \( x = 3 + 2\sqrt{2} \), we will rationalize the denominator. Here are the steps:

### Step 1: Write the expression for \( \frac{1}{x} \)

We start with:
\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \]

### Step 2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 3 - 2\sqrt{2} \):
\[ \frac{1}{3 + 2\sqrt{2}} \cdot \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]

### Step 3: Simplify the denominator using the difference of squares

Now we simplify the denominator using the difference of squares formula \( a^2 - b^2 \):
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 \]
Calculating each part:
\[ 3^2 = 9 \quad \text{and} \quad (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]
So, we have:
\[ 9 - 8 = 1 \]

### Step 4: Write the final expression

Now substituting back into our expression:
\[ \frac{3 - 2\sqrt{2}}{1} = 3 - 2\sqrt{2} \]

### Final Answer

Thus, the value of \( \frac{1}{x} \) is:
\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

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