Final answer:
The multiplicative inverse of a non-zero number a is 1/a, which can also be expressed by raising a to the power of -1 (a^-1). This inverse, when multiplied with the original number, results in a product of 1. It's similar to how negative exponents indicate reciprocals in exponentiation.
Step-by-step explanation:
When you have a non-zero number a, its multiplicative inverse is the number that, when multiplied by a, gives the product of 1. This is also known as the reciprocal of the number. So, if a ≠ 0, the multiplicative inverse of a is 1/a. In the context of exponents, this is similar to raising a to the power of -1 (a-1).
For example, let's say you have a scalar a and a vector Ả = AxÎ + AyĴ + AzÊ, if a is set to -1, then the antiparallel vector would be -Ả = -AxÎ - AyĴ - Azk. Multiplying the scalar by its multiplicative inverse (in this case -1), you’ll get the scalar's original magnitude but in the opposite direction.
Furthermore, dealing with powers, if you have a term with a power (such as a2), you can neutralize the power by raising the term to an inverse power. This principle is reflected in exponent rules, particularly when dealing with negative exponents, where an exponent of -n denotes a reciprocal, as described by equation A.9 where 1/xn = x-n.
It is important to remember that a multiplicative inverse exists only for non-zero numbers, as division by zero is undefined.