Convert `(4-isqrt(3))/(4+isqrt(3))` in the form of `a+ib`.

Explanation

To convert the expression \(\frac{4 - i\sqrt{3}}{4 + i\sqrt{3}}\) into the form \(a + ib\), we will follow these steps:

### Step 1: Rationalize the denominator
To eliminate the imaginary part in the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(4 - i\sqrt{3}\).

\[ \frac{4 - i\sqrt{3}}{4 + i\sqrt{3}} \cdot \frac{4 - i\sqrt{3}}{4 - i\sqrt{3}} = \frac{(4 - i\sqrt{3})^2}{(4 + i\sqrt{3})(4 - i\sqrt{3})} \]

### Step 2: Expand the numerator
Now, we will expand the numerator \((4 - i\sqrt{3})^2\):

\[ (4 - i\sqrt{3})^2 = 4^2 - 2 \cdot 4 \cdot i\sqrt{3} + (i\sqrt{3})^2 = 16 - 8i\sqrt{3} - 3 \]

This simplifies to:

\[ 16 - 3 - 8i\sqrt{3} = 13 - 8i\sqrt{3} \]

### Step 3: Expand the denominator
Next, we will expand the denominator \((4 + i\sqrt{3})(4 - i\sqrt{3})\):

\[ (4 + i\sqrt{3})(4 - i\sqrt{3}) = 4^2 - (i\sqrt{3})^2 = 16 - (-3) = 16 + 3 = 19 \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{13 - 8i\sqrt{3}}{19} \]

### Step 5: Separate into real and imaginary parts
We can separate this into real and imaginary parts:

\[ \frac{13}{19} - \frac{8\sqrt{3}}{19}i \]

### Final Result
Thus, the expression in the form \(a + ib\) is:

\[ \frac{13}{19} - \frac{8\sqrt{3}}{19}i \]