(3+4i)/(4-5i)

To find the multiplicative inverse, all you need to do is flip the function so the product of the number and its inverse = 1

Then the inverse is:

(4-5i)/(3+4i)

Now let us simplify:

Multiply and divide by (3-4i)

==> (3-4i)(4-5i)/ (3+4i)(3-4i)

==> (12- 15i - 16i - 20)/(3^2 - (4i)^2]

==> (-8 - 29i)/(9+ 16) = (-8-29i)/ 25

==> -8/25 - (29/25)i

Let us simplify (3+4i/(4-5i).

(3+4i)/(4-5i) = (3+4i)(4+5i)/(4-5i)(4+5i) = (12-20 +15i+16i)/{16+25+20i-20i}

=(-8+31i)/(41) = (-8/41) +(31/41)i

The identity element of complex number is 1.

The multiplicative inverse of x+iy = 1/(x+iy) = (x-iy)/ (x^2+y^2)

Therefore the multiplicative inverse of (-8/41)+(31/41)i = -(8/41)-(31/41)]/[(8/41)^2 + 31/41)^2]

= - 41(8+31i)/(8^2+31^2)

=-41(8+31i)/(1025)

The multiplicative inverse of the given ratio is:

(4-5i)/(3+4i)

Now, because it is not allowed to have a complex number at denominator, we'll multiply the ratio by the conjugate of (3+4i).

(4-5i)/(3+4i) = (4-5i)*(3-4i)/(3+4i)*(3-4i)

We'll remove the brackets:

(4-5i)*(3-4i) = 12 - 16i - 15i - 20 = -8-31i = -(8+31i)

(3+4i)*(3-4i) = (3)^2 - (4i)^2 = 9 + 16 = 25

**(4-5i)/(3+4i) = -(8+31i)/25**

**The multiplicative inverse is:**

**-8/25 - (31/25)*i**