Page 36 - Start Up Mathematics_8 (Non CCE)

2 Exponents (Powers)

When a rational number x (x ≠ 0) is repeatedly multiplied by itself n times (where n is a natural number), it
n
is written as x × x × x × ... n times or x . This notation or style of writing is called exponential notation or
power notation. It is also known as the nth power of x or x raised to the power n. The rational number x is
called the base and the natural number n is called the exponent or the index.

Laws of Exponents of Rational Numbers

If x and y are two non-zero rational numbers and m, n are natural numbers, then
m
n
mn
n m
m n
m
n
1. (x) × (x) = (x) m + n 2. (x) ÷ (x) = (x) m – n (where m > n) 3. (x ) = (x) = (x )
.
n
1
n
4. (xy) = (x) (y) n 5.  x  n = x () n n 6. (x) = x
 
y
  y ()
0
7. (x) = 1 (zero-exponent property)
Negative integral exponents of a rational number
2
You already know that, (10) = 10 × 10 = 100
1
(10) = 10
0
(10) = 1
Did you notice a pattern emerging here? The value decreases by one-tenth when the exponent decreases by one.
0
So, ()10 − 1 = 10 ÷ 10 110=÷ = 1
10
()10 − 2 = ()10 − 1 ÷ 10 = 1 ÷ 10 = 1 × 1 = 1 = 1
10 10 10 100 ()10 2

()10 − 3 = ()10 − 2 ÷ 10 = 1 ÷ 10 = 1 × 1 = 1 = 1 and so on.
100 100 10 , 1 000 ()10 3

p  p  − n 1  q  n
If is a rational number and n is any positive integer, then   = =  p  . In other words, if
q  q   p  n  
 q 
 
1
–n
–n
x (x ≠ 0) is any rational number and n is any positive integer, then (x) = . Here, (x) is called the
() x n
n
reciprocal of (x) .
p
Example 1: Express each of the following in the form .
q

− 
(a) (4) –4 (b)  3  −4 (c) 1 (d)  5  − 7 ×  8  − 4 (NCERT)
 
 

 5  2 () − 3  8   5 