Answer:
1) The multiplicative inverse of [tex]8[/tex] is [tex]\frac{1}{8}[/tex].
2) The multiplicative inverse of [tex]8\cdot h - 40[/tex] is [tex]\frac{1}{8\cdot h - 40}[/tex].
Step-by-step explanation:
Mathematically, let [tex]w[/tex] and [tex]v[/tex] real numbers. [tex]w[/tex] is the multiplicative inverse if and only if [tex]v\cdot w = 1[/tex]. Now we proceed to determine the multiplicative inverse of each number:
1) [tex]v = 8[/tex]
(i) [tex]v\cdot w = 1[/tex] Definition of multiplicative inverse
(ii) [tex]v = 8[/tex] Given
(iii) [tex]8\cdot w = 1[/tex] (ii) in (i)
(iv) [tex]w\cdot (8\cdot 8^{-1}) = 8^{-1}\cdot 1[/tex] Compatibility with multiplication/Associative and commutative properties
(v) [tex]w = 8^{-1}[/tex] Existence of multiplicative inverse/Modulative property
(vi) [tex]w = \frac{1}{8}[/tex] Definition of division/Result
The multiplicative inverse of [tex]8[/tex] is [tex]\frac{1}{8}[/tex].
2) [tex]v = 8\cdot h - 40[/tex]
(i) [tex]v\cdot w = 1[/tex] Definition of multiplicative inverse
(ii) [tex]v = 8\cdot h - 40[/tex] Given
(iii) [tex](8\cdot h - 40)\cdot w = 1[/tex] (ii) in (i)
(iv) [tex]w\cdot [(8\cdot h - 40)\cdot (8\cdot h-40)^{-1}] = (8\cdot h - 40)^{-1}\cdot 1[/tex] Compatibility with multiplication/Associative and commutative properties
(v) [tex]w = (8\cdot h-40)^{-1}[/tex] Existence of multiplicative inverse/Modulative property
(vi) [tex]w = \frac{1}{8\cdot h-40}[/tex] Definition of division/Result
The multiplicative inverse of [tex]8\cdot h - 40[/tex] is [tex]\frac{1}{8\cdot h - 40}[/tex].
