Multiplicative inverse, also known as reciprocal, of a number is a number that, when multiplied by the original number, gives the product of 1. In other words, if we have a number 'x', then its multiplicative inverse is denoted as '1/x'.

(a) Multiplicative Inverse of 1/6:
To find the multiplicative inverse of 1/6, we need to find a number that, when multiplied by 1/6, gives the product of 1.

Let's assume the multiplicative inverse of 1/6 is 'y'. Therefore, we have the equation: (1/6) * y = 1.

To solve for 'y', we can multiply both sides of the equation by 6 to eliminate the fraction:
(1/6) * y * 6 = 1 * 6
y = 6

So, the multiplicative inverse of 1/6 is 6.

(b) Multiplicative Inverse of -3/8:
To find the multiplicative inverse of -3/8, we need to find a number that, when multiplied by -3/8, gives the product of 1.

Let's assume the multiplicative inverse of -3/8 is 'y'. Therefore, we have the equation: (-3/8) * y = 1.

To solve for 'y', we can multiply both sides of the equation by -8 to eliminate the fraction:
(-3/8) * y * (-8) = 1 * (-8)
y = -8/3

So, the multiplicative inverse of -3/8 is -8/3.

(c) Multiplicative Inverse of 4/19:
To find the multiplicative inverse of 4/19, we need to find a number that, when multiplied by 4/19, gives the product of 1.

Let's assume the multiplicative inverse of 4/19 is 'y'. Therefore, we have the equation: (4/19) * y = 1.

To solve for 'y', we can multiply both sides of the equation by 19 to eliminate the fraction:
(4/19) * y * 19 = 1 * 19
y = 19/4

So, the multiplicative inverse of 4/19 is 19/4.

In summary, the multiplicative inverse of 1/6 is 6, the multiplicative inverse of -3/8 is -8/3, and the multiplicative inverse of 4/19 is 19/4.

A