# Find the multiplicative inverse of the complex number 4 - 2i.

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We have to find the multiplicative inverse of 4 - 2i.

Now the multiplicative inverse of any number X is defined as X^-1, so that X*X^-1 = 1.

For 4 - 2i, let the multiplicative inverse be M.

Now M* (4 - 2i) = 1

=> M = 1/ (4 -2i)

=> M = (4 +2i)/ ( 4- 2i)( 4 +2i)

=> M = (4 + 2i) / [4^2 - (2i)^2]

=> M = ( 4 + 2i) / [ 16 + 4]

=> M = (4 + 2i) / 20

=> M = 4/20 + 2i /20

=> M = 1/5 + i/10

**Therefore the multiplicative inverse of 4 - 2i is 1/5 + i/10.**

To find the multiplicative inverse of the complex number.

We know the multiplication identity element of the complex number (x+yi) = e(x+iy) .

Therefore xe = x and and xi*e = xi. Therefore e = 1.

Therfore if the multiplicative inverse of 4-2i is x+yi, then

(4-2i)(x+yi) = 1.

4x+4yi-2xi -2yi^2 = 1+0*i

4x +(4y-2x)i +2y = 1+0*i.

(4x+2y) +(4y-2x)i = 1+0*i.

Equate imaginary parts and equate real parts both sides:

Imaginary parts: 4y -2x = 0.....(1)

Real Parts: 4x+2y = 1.....(2)

From (1): 4y- 2x = 0.

x= 2y. Substitute x= 2y in eq (2):

4(2y) +2y = 1

10y = 1.

y = 1/10.

x =2y= 2/10.

Therefore (4-2i) has the multiplicative inverse 2/10+i/10.

4y + 2(2y) =

The multiplicative inverse is get when we multiply the given complex number by the inverse number we'll get the result 1.

We'll note the inverse as x:

(4-2i)*x = 1

We'll divide by (4-2i) both sides:

x = 1/(4-2i)

We are not allowed to keep a complex number to denominator, so we'll multiply the ratio by the conjugate of the complex number:

x = (4+2i)/(4-2i)(4+2i)

We'll re-write the denominator as a difference of squares:

(4-2i)(4+2i) = 4^2 - (2i)^2

(4-2i)(4+2i) = 16 - 4i^2

But i^2 = -1:

(4-2i)(4+2i) = 16 + 4

(4-2i)(4+2i) = 20

We'll re-write x:

x = (4+2i)/20

x = 4/20 + 2i/20

We'll simplify and we'll get:

x = 1/5 + i/10

**The multiplicative inverse of the complex number 4 - 2i is 1/5 + i/10.**