Rationalise the denominator of `(1)/(8+4sqrt(2))`

Answer

To rationalize the denominator of the expression \(\frac{1}{8 + 4\sqrt{2}}\), we will follow these steps:

### Step 1: Identify the conjugate
The denominator is \(8 + 4\sqrt{2}\). The conjugate of this expression is \(8 - 4\sqrt{2}\).

### Step 2: Multiply numerator and denominator by the conjugate
We multiply both the numerator and the denominator by the conjugate:
\[ \frac{1}{8 + 4\sqrt{2}} \times \frac{8 - 4\sqrt{2}}{8 - 4\sqrt{2}} = \frac{8 - 4\sqrt{2}}{(8 + 4\sqrt{2})(8 - 4\sqrt{2})} \]

### Step 3: Simplify the denominator using the difference of squares
Using the difference of squares formula \(a^2 - b^2\), where \(a = 8\) and \(b = 4\sqrt{2}\):
\[ (8 + 4\sqrt{2})(8 - 4\sqrt{2}) = 8^2 - (4\sqrt{2})^2 \]
Calculating \(8^2\):
\[ 8^2 = 64 \]
Calculating \((4\sqrt{2})^2\):
\[ (4\sqrt{2})^2 = 16 \cdot 2 = 32 \]
So, the denominator becomes:
\[ 64 - 32 = 32 \]

### Step 4: Write the final expression
Now, we can write the expression:
\[ \frac{8 - 4\sqrt{2}}{32} \]

### Step 5: Simplify the fraction
We can simplify the fraction by dividing both terms in the numerator by 32:
\[ \frac{8}{32} - \frac{4\sqrt{2}}{32} = \frac{1}{4} - \frac{\sqrt{2}}{8} \]

Thus, the rationalized form of \(\frac{1}{8 + 4\sqrt{2}}\) is:
\[ \frac{1}{4} - \frac{\sqrt{2}}{8} \]