The multiplicative inverse of an element of a set is another element of this set such that the product of the two elements is a multiplicative identity. In the given set of complex numbers, the multiplicative identity is 1. This is because multiplication by 1 does not change any complex number:
(a + bi)*1 = a + bi for any a and b.
Then, to find multiplicative inverse of 6 - 3i, we need to find the complex number such that the product of 6 - 3i and that number is 1. We can find it by performing division of 1 by 6 - 3i:
1/(6 - 3i) = ?
To find the result, multiply the numerator and denominator of the fraction above by the complex conjugate of 6 - 3i: 6 + 3i.
1/(6 - 3i) = (6+3i)/[(6-3i)(6+3i)]
In the denominator, the product of the two complex conjugates results in a real number:
(6-3i)(6+3i) = 36 - 9i^2 = 36 + 9 = 45. (Keep in mind that i^2 = -1.)
1/(6 - 3i) = (6 + 3i)/45 = 2/15 + (1/15)i.
The multiplicative inverse of 6 - 3i is 2/15 + (1/15)i.
This could be double-checked by performing the multiplication:
(6-3i)(2/15 + (1/15)i) = 12/15 - (6/15)i + (6/15)i - (3/15)i^2 = 4/5+ 1/5 = 1, as expected.