Find the multiplicative inverse of the complex number √5 + 3i.

To find the multiplicative inverse of a complex number, we need to divide 1 by the given complex number.

Let's start by representing the given complex number as √5 + 3i.

To find the multiplicative inverse, we need to multiply the numerator and denominator by the conjugate of the given complex number.

The conjugate of √5 + 3i is √5 - 3i.

So, multiplying the numerator and denominator by √5 - 3i, we get:

(1 * (√5 - 3i)) / ((√5 + 3i) * (√5 - 3i))

Simplifying this expression, we have:

(√5 - 3i) / ((√5)^2 - (3i)^2)

Simplifying further, we have:

(√5 - 3i) / (5 - 9i^2)

Since i^2 is equal to -1, we can substitute it in the expression:

(√5 - 3i) / (5 - 9(-1))

Simplifying this expression, we have:

(√5 - 3i) / (5 + 9)

Finally, simplifying further, we get:

(√5 - 3i) / 14

Therefore, the multiplicative inverse of the complex number √5 + 3i is (√5 - 3i) / 14.