Rationalise the denominator of `(1)/(7+4sqrt(3))`.

Explanation

To rationalize the denominator of the expression \( \frac{1}{7 + 4\sqrt{3}} \), we can follow these steps:

### Step 1: Identify the Conjugate
The denominator is \( 7 + 4\sqrt{3} \). The conjugate of this expression is \( 7 - 4\sqrt{3} \).

### Step 2: Multiply by the Conjugate
Multiply both the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{7 + 4\sqrt{3}} \times \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} = \frac{1 \cdot (7 - 4\sqrt{3})}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

### Step 3: Simplify the Numerator
The numerator simplifies to:

\[ 7 - 4\sqrt{3} \]

### Step 4: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula \( a^2 - b^2 \):

\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 \]

Calculating \( 7^2 \) and \( (4\sqrt{3})^2 \):

\[ 7^2 = 49 \]
\[ (4\sqrt{3})^2 = 16 \cdot 3 = 48 \]

So, the denominator becomes:

\[ 49 - 48 = 1 \]

### Step 5: Write the Final Expression
Now we combine the simplified numerator and denominator:

\[ \frac{7 - 4\sqrt{3}}{1} = 7 - 4\sqrt{3} \]

Thus, the rationalized form of \( \frac{1}{7 + 4\sqrt{3}} \) is:

\[ 7 - 4\sqrt{3} \]

### Summary of Steps
1. Identify the conjugate of the denominator.
2. Multiply the numerator and denominator by the conjugate.
3. Simplify the numerator.
4. Simplify the denominator using the difference of squares.
5. Combine the results into the final expression.

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