Page 33 - ICSE Math 8

2 Exponents and Powers

Key Concepts

• Laws of Exponents of Rational Numbers • Negative Integral Exponents

When a rational number x (x ≠ 0) is repeatedly multiplied by itself n times (where n is a natural number), it is
n
written as x × x × x × ... n times or x . This notation is called exponential notation or power notation. It is read
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 be any integers, then the following Laws Statement Name of the law
laws of integral exponents can be defined as Law I (x) × (x) = (x) m + n Law of products
m
n
shown alongside.
Law II (x) m m – n , where m > n Law of quotients
n = (x)
Negative Integral Exponents (x)
n m
mn
m n
0
1
We already know that, 10 = 1, 10 = 10, Law III (x ) = (x) = (x ) Law of powers
2
3
10 = 100, 10 = 1,000 and so on. Law IV (xy) = (x) × (y) n
n
n
Did you notice a pattern emerging here?
The value increases by ten when the exponent Law V  x  n = x () n
increases by one.   y () n
y
 
1000 100 10
Also, = 100, = 10, = 1
10 10 10 (x) = 1 Zero exponent
0
In exponent notation these results can be
written as follows:
3
2
10 3 10 10 2 10 10 1 10 1
2
0
1
2
0
1
= 10 or 10 = , = 10 or 10 = , = 10 = 1 or 1 = 10 =
10 10 10 10 10 10
The above results show exhibit a pattern that as the exponents of 10 decreased by 1 the value becomes one
tenth of the previous value. So if the same pattern is continued, we must have
1
–1
10 =
10 1 1 1 1
–2
–1
10 = (10 ) ÷ 10 = ÷ 10 = × = = 1
10 10 10 100 10 2
–2
–3
10 = (10 ) ÷ 10 = 1 ÷ 10 = 1 × 1 = 1 = 1 and so on.
100 100 10 1000 10 3
This suggests us the following definition for negative integral exponents of a non-zero rational number.
p  p  − n 1  q  n q () n
If q is a rational number and n is any positive integer, then   q  = n =   p   or p () n . In other words,

p 
  q 
 
–n
–n
if x (x ≠ 0) is any rational number and n is any positive integer, then (x) = () x 1 n . Here, (x) is called the
n
reciprocal of (x) .
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