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Exponents

Teacher: You all know, that when a number is added repeatedly many times, instead of adding
the number each time, we use multiplication to represent it.

For example: (a) 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 8 × 3 (read as 8 times 3)
(b) 7 + 7 + 7 + 7 + 7 + 7 = 6 × 7 (read as 6 times 7)

But what about when a number is multiplied by itself repeatedly many times, how do we write it?
The repeated multiplication of any number gives an idea of powers or exponents.
Let us observe 2 × 2 × 2 × 2 × 2 × 2, here 2 is multiplied itself 6 times.

6
It can be written as 2 . Here, 2 is called the base and 6 is called the exponent or power and it is
read as ‘2 raised to the power 6’ or simply as ‘sixth power of 2’.
2 is the short notation for 2 × 2 × 2 × 2 × 2 × 2. 2 6 Exponent
6
6
Thus, 2 × 2 × 2 × 2 × 2 × 2 = 2 = 64 [2 raised to the power 6] Base
6
Thus, we can say that 2 is the exponential form of 64.
5
Similarly, (a) 3 × 3 × 3 × 3 × 3 = 3 = 243 [3 raised to the power 5]
Remember
5
3 is the exponential form of 243. The exponent is the
4
(b) 10 × 10 × 10 × 10 = 10 = 10000 [10 raised to the power 4] number of times a number
4
10 is the exponential form of 10000. is used in a multiplication.

Enrichment
We can express a number into their expanded form using exponential notation. As,
37563 = 3 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 3
3
2
4
= 3 × 10 + 7 × 10 + 5 × 10 + 6 × 10 + 3
Express the following numbers into their expanded form using exponents.
(a) 65841 (b) 98754 (c) 8007465
Base as Negative Integer

We can also extend this way of writing when the base is a negative integer. Let us assume the
following examples.
3
(a) (–4) × (−4) × (−4) = (−4) [(–4) raised to the power 3]
Or (–4) × (−4) × (−4) = −64 (Q The product of negative integers when multiplied odd number
of times, the result is a negative integer.)

3
\ (−4) = (–4) × (−4) × (−4) = −64 = –(4) 3
3
So, (–4) is the exponential form of (–64).
4
(b) (−5) = (−5) × (−5) × (−5) × (−5) = 625 = 5 4
n
n
n
n
Thus, in general, (−x) = x , if n is an even number and (−x) = −x , if n is an odd number.
If x =1, then we have
n
 − ( ) = 1 n = 1 if is an even number
n
1
,

 n
n
,
 − ( ) =−1 1 n =−1if is an odd number

79 Exponents and Powers