Question

Find the multiplicative inverse of the product of complex
numbers :
3 + 4i, 5 - 12i.​

Answer 1

Answer:

$\large{\bold{\frac{(23-14i)}{725}}}$

Step-by-step explanation:

Multiplicative of a number is a number, whose product with the original number gives product as 1.

Given No

( 3 + 4i ) x ( 5 - 2i )

Now the inverse is

$\frac{1}{(3+4i)(5-2i)}\\\\=\frac{1}{23+14i}\\\\Rationalise\;denominator\\\\=\frac{1}{(23+14i)}\times\frac{(23-14i)}{(23-14i)}\\\\=\frac{(23-14i)}{23^2+14^2}\\\\=\frac{(23-14i)}{725}$

HOPE IT HELPS

Answer 2

$let \sqrt{ - 5 + 12i} = a + ib$

$- 5 + 12i = (a + ib {)}^{2} = {a}^{2} - {b}^{2} + 2iab$

$We \: will \: get \: {a}^{2} - {b}^{2} = - 5, \: 2ab = 12$

${a}^{2} - \frac{36}{ {a}^{2} } = - 5.......b = \frac{6}{a}$

${a}^{4} + 5 {a}^{2} - 36 = 0$

$( {a}^{2} - 4)( {a}^{2} + 9) = 0$

$a = ±2 \: and \: ±3$

$we \: have \: a = 2 \: and \: b = 3$

$a = - 2 \: and \: b = - 3$

$b = \frac{6}{a} = ±3 so \sqrt{ - 5 + 12i}$

$= ±(2 + 3i)$

hope it's helps you ❤️