Page 34 - ICSE Math 8

Properties of negative integral exponents

The laws of exponents hold good for negative integral exponents of a rational numbers. Let us understand it
with the help of the following examples.
p
Example 1: Express each of the following in the form q .

− 
(a) (4) –4 (b)  3  −4 (c) 1

 5  2 () − 3
–4
Solution: (a) (4) = 1 = 1 = 1    x − n = 1 n  
×××
4 () 4 4444 256  x 
n 

− 
 3 −4 1 1   x  n x 
(b)   = =     = 
 5   3 4 (− 3) 4     y 
n
y
− 

  4
 5  5 ()
()5 4 5555 625
×××
= = =
−
) ( 3 × −
) ( 3
−
( 3 ) 4 ( 3 × − ) ( 3 × − ) 81
1 1  1 
3
(c) = = (2) = 2 × 2 × 2 = 8   x − n = n 
2 () − 3 1  x 
2 () 3
Example 2: Write each of the following as a rational number with positive exponent.
–5
–7
(a) {(3) ÷ (3) –10 } × (3) (b)  −  1  −5 ×   −  1  −7

 5   5 
–7
Solution: (a) {(3) ÷ (3) –10 } × (3) –5

=  1 ÷ 1  × 1  ()x − n = 1 n 

10 

 3 () 7 3 ()  3 () 5  ()x 
10
 1 3 ()  1  3 ()  1
10




=  ×  × =   ×
7
  3 () 7 1  3 () 5   3 ()  3 () 5


−
2
= 3 ( 10 7− ) × 1 = () 3 1 = 3 () 3 − 5 = 3 ()   ()x m = ()x m − n 
3 ×


3 () 5 3 () 5  ()x n 

n
1
1 1 () 2   2  ()x n  x 

= = =     =   
3
3 () 2 3 () 2     ()y n  y 

 −  1 −5  −  1 −7 1 1 1 1
(b)   ×   = × = ×
1
1
 5   5   − 1 5  − 1 7 − () 5 − () 7
    5 7
 5   5  5 () 5 ()
= ()5 5 × ()5 7 = ()5 5 + 7 = ()5 12
−
−
−
()1 5 ()1 7 ()1 5 + 7 () 1 12
−
()5 12 12
= = (5) { (negative number) even power = positive number}
1
 1  2
Example 3: By what number should   be multiplied to get the product as 15?
 3 
Solution: Let the required number be x.
 1  2  1  2
  × x = 15 ⇒ x ×   = 15 ( Multiplication is commutative)
 3   3 
22