Write the multiplicative Inverse of `1+4sqrt3i`

Answer

To find the multiplicative inverse of the complex number \(1 + 4\sqrt{3}i\), we can follow these steps:

### Step 1: Define the multiplicative inverse
Let \(x\) be the multiplicative inverse of \(1 + 4\sqrt{3}i\). By definition, this means:
\[ x(1 + 4\sqrt{3}i) = 1 \]
Thus, we can express \(x\) as:
\[ x = \frac{1}{1 + 4\sqrt{3}i} \]

### Step 2: Rationalize the denominator
To simplify \(\frac{1}{1 + 4\sqrt{3}i}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 - 4\sqrt{3}i\):
\[ x = \frac{1 \cdot (1 - 4\sqrt{3}i)}{(1 + 4\sqrt{3}i)(1 - 4\sqrt{3}i)} \]

### Step 3: Calculate the denominator
Using the difference of squares formula \(A^2 - B^2\), where \(A = 1\) and \(B = 4\sqrt{3}i\):
\[ (1 + 4\sqrt{3}i)(1 - 4\sqrt{3}i) = 1^2 - (4\sqrt{3}i)^2 \]
Calculating this gives:
\[ 1 - (16 \cdot 3)(-1) = 1 - (-48) = 1 + 48 = 49 \]

### Step 4: Substitute back into the expression for \(x\)
Now substituting back, we have:
\[ x = \frac{1 - 4\sqrt{3}i}{49} \]

### Step 5: Split the fraction
This can be expressed as:
\[ x = \frac{1}{49} - \frac{4\sqrt{3}}{49}i \]

### Final Answer
Thus, the multiplicative inverse of \(1 + 4\sqrt{3}i\) is:
\[ \frac{1}{49} - \frac{4\sqrt{3}}{49}i \]

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