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Multiplicative Inverse or Reciprocals

1
1
−
We know that reciprocal of an integer a is a = a . Similarly, the reciprocal of a rational number
 p    p −1  q
1
  is  p  or   or   .
q

q


p
 
 q   1 5 1
5
–5
For example, the reciprocal of (3) =   = 3 5 = 3 .

3
Similarly, the reciprocal of  4 3 = 1 3 =  4 − 3 .
 
 
5
5


4

 
5

The reciprocal is also known as the multiplicative inverse.
Example 2: Find the reciprocal of
2
(a) (–2) 5 (b)    − 3

5
1 1
5
Solution: (a) Reciprocal of (–2) = =
− ( 2) 5 − 32
8
2
2
(b) Reciprocal of    − 3 = 1 − 3 =    3 = 125
5

5


2
 
5

(
1
0
Example 3: Simplify: 4 + ) ÷ 2 0 0    − 2

2
1
1
1
(
0
0
Solution: 4 + ) ÷ 2 0 0  1 − 2 = ( 11) ÷    − 2 = 2 ×    =× 1 = 1     Q Q Reciprocal of    − 2 =  1 2 
1
0
+
  
 


2
2

2


2
2
4
4
 
(
1 
−
3
Example 4: Simplify: 4 × 2 ) ÷    10  − 2
2
1 
1 

1 
1 
(
3
2
−
Solution: 4 × 2 ) ÷    10 − 2 = 16 × 2  ÷    10 − 2 =    16 × 1  ÷  10  − 2

3 




8

 1  2 2 1
= 2 ×  10  = 100 = 50

q
p
r
 x  r  x  p  x  q
Example 5: Show that   x  ×  x  ×  x  = 1 . Think and Answer
r 
q 
p 


–3
1. Find the value of 6 .
q
r
 x  r  x  p  x  q –20
p
Solution: LHS =   x  ×   x  ×   x  2. Find the multiplicative inverse of 100 .
q 
p 
r 
)
) × (x
) × (x
= (x p – q r q – r p r – p q
mn
m n
= x pr – qr × x qp – rp × x rq – pq [Q(a ) = a ]
n
= x pr – qr + qp – rp + rq – pq [Qa × a = a m + n ]
m
= x = 1 (RHS) [Qa = 1]
0
0
267 Exponents and Powers