Question

The multiplicative inverse of 8 and 8h - 40

Answer 1

1) The multiplicative inverse of
`8` is
`(1)/(8)`.

2) The multiplicative inverse of
`8\cdot h - 40` is
`(1)/(8\cdot h - 40)`.

Explanation:

Mathematically, let
`w` and
`v` real numbers.
`w` is the multiplicative inverse if and only if
`v\cdot w = 1`. Now we proceed to determine the multiplicative inverse of each number:

1)
`v = 8`

(i)
`v\cdot w = 1` Definition of multiplicative inverse

(ii)
`v = 8` Given

(iii)
`8\cdot w = 1` (ii) in (i)

(iv)
`w\cdot (8\cdot 8^(-1)) = 8^(-1)\cdot 1` Compatibility with multiplication/Associative and commutative properties

(v)
`w = 8^(-1)` Existence of multiplicative inverse/Modulative property

(vi)
`w = (1)/(8)` Definition of division/Result

The multiplicative inverse of
`8` is
`(1)/(8)`.

2)
`v = 8\cdot h - 40`

(i)
`v\cdot w = 1` Definition of multiplicative inverse

(ii)
`v = 8\cdot h - 40` Given

(iii)
`(8\cdot h - 40)\cdot w = 1` (ii) in (i)

(iv)
`w\cdot [(8\cdot h - 40)\cdot (8\cdot h-40)^(-1)] = (8\cdot h - 40)^(-1)\cdot 1` Compatibility with multiplication/Associative and commutative properties

(v)
`w = (8\cdot h-40)^(-1)` Existence of multiplicative inverse/Modulative property

(vi)
`w = (1)/(8\cdot h-40)` Definition of division/Result

The multiplicative inverse of
`8\cdot h - 40` is
`(1)/(8\cdot h - 40)`.