Page 84 - Start Up Mathematics_7

5 Exponents and Powers

Do you know that the distance between the sun and the
earth is 150,000,000,000 m and the distance between the
sun and saturn is 1,426,000,000,000 m. Can you tell which
of the two planets earth or saturn is nearer to the sun? Did
you find it difficult to read these large numbers? To make
it convenient to read and understand large numbers we
use exponents.
Exponents help us to read, understand and compare very large and very small numbers like
population of countries, distance between planets, size of atoms etc.

Exponents
7
Large number like 10,000,000 can be written in shorter form as 10 . Clearly, 10,000,000 =
7
10 × 10 × 10 × 10 × 10 × 10 × 10 = 10 . It is read as 10 raised to the power 7 or seventh power
7
7
of 10. 10 is called the exponential form of 10,000,000. In 10 , base is 10 and exponent is 7.
8
Also, 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 , here base is Do you know?
2 and exponent is 8. Some powers have special names. For (i) (1) any power = 1
2
example, 5 , which is 5 raised to the power 2, also read as (ii) (–1) odd number = –1
‘5 squared’ means 5 is to be multiplied by itself two times (iii) (–1) even number = 1
3
and 5 , which is 5 raised to the power 3, also read as ‘5
cubed’ means 5 is to be multiplied by itself three times. Verify these results for different powers.
3
7
3
Example 1: Find the value of: (a) 2 (b) 7 (c) 11 (d) 5 4
7
3
Solution: (a) 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 (b) 7 = 7 × 7 × 7 = 343
4
3
(c) 11 = 11 × 11 × 11 = 1,331 (d) 5 = 5 × 5 × 5 × 5 = 625
Example 2: Express the following in exponential form:
(a) 2 × 2 × 2 × 2 (b) x × x × x × x × x
(c) 3 × 3 × 3 × 3 × b × b × b (d) a × a × c × c × c × d × d

4
Solution: (a) 2 × 2 × 2 × 2 = 2 (b) x × x × x × x × x = x 5
4 3
2 3 2
(c) 3 × 3 × 3 × 3 × b × b × b = 3 b (d) a × a × c × c × c × d × d = a c d
Example 3: Express each of the following numbers using exponential notation:
(a) 1,024 (b) 243 (c) 15,625

Solution: (a) 1,024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 10
(b) 243 = 3 × 3 × 3 × 3 × 3 = 3 5
(c) 15,625 = 5 × 5 × 5 × 5 × 5 × 5 = 5 6