​ Find the multiplicative inverse of each of the following: (1+i)(1+2i)/(1+3i) ​

Question

Find the multiplicative inverse of each of the following:

\(\frac{(1+i)(1+2i)}{(1+3i)}\)

(1+i)(1+2i)/(1+3i)

Answer 1

Given: \(\frac{(1+i)(1+2i)}{(1+3i)}\)

To find: Multiplicative inverse

We know that,

Multiplicative Inverse of z = z^-1

We solve the above equation

Now, we rationalize the above by multiplying and divide by the conjugate of (-1 + 3i)

Now, we know that,

(a + b)(a – b) = (a² – b²)

So, eq. (i) become

Hence, Multiplicative inverse of

\(\frac{(1+i)(1+2i)}{(1+3i)}\) \(=\frac{4}{5}-\frac{3}{5}i\)