48 Basic Engineering Mathematics

Problem 3. Evaluate 3^3 × 22

33 × 22 = 3 × 3 × 3 × 2 × 2

= 27 × 4

= 108

7.2.2 Square roots

When a number is multiplied by itself the product is

called a square.

For example, the square of 3 is 3× 3 = 32 =9.

A square root is the reverse process; i.e., the value of the

base which when multiplied by itself gives the number;

i.e., the square root of 9 is 3.

The symbol

√

√ is used to denote a square root. Thus,

9 =3. Similarly,

√

4 =2and

√

25 =5.

Because− 3 ×− 3 = 9 ,

√

9alsoequals−3. Thus,

√

9 =

+3or−3 which is usually written as

√

9 =±3. Simi-

larly,

√

16 =±4and

√

36 =±6.

The square root of, say, 9 may also be written in index

form as 9

(^12)
9
1
(^2) ≡
√
9 =± 3
Problem 4. Evaluate
32 × 23 ×
√
36
√
16 × 4
taking only
positive square roots
32 × 23 ×
√
36
√
16 × 4

3 × 3 × 2 × 2 × 2 × 6
4 × 4

9 × 8 × 6
16

9 × 1 × 6
2

9 × 1 × 3
1
by cancelling
= 27
Problem 5. Evaluate
104 ×
√
100
103
taking the
positive square root only
104 ×
√
100
103

10 × 10 × 10 × 10 × 10
10 × 10 × 10

1 × 1 × 1 × 10 × 10
1 × 1 × 1
by cancelling

100
1
= 100
Now try the following Practice Exercise
PracticeExercise 29 Powersand roots
(answers on page 342)
Evaluate the following without the aid of a calcu-
lator.

1. 3^3 2. 2^7


2. 10^5 4. 2^4 × 32 × 2 ÷ 3


3. Change 16 to 6. 25

1

2

index form.

7. 64

1

(^2) 8.
105
103
9.
102 × 103
105
10.
25 × 64
1
(^2) × 32
√
144 × 3
taking positive
square roots only.

7.3 Laws of indices

There are six laws of indices.

(1) From earlier, 2^2 × 23 =( 2 × 2 )×( 2 × 2 × 2 )

= 32

= 25

Hence, 22 × 23 = 25

or 22 × 23 = 22 +^3

This is the first law of indices, which demonstrates

thatwhen multiplying two or more numbers

having the same base, the indices are added.

(2)

25

23

=

2 × 2 × 2 × 2 × 2

2 × 2 × 2

=

1 × 1 × 1 × 2 × 2

1 × 1 × 1

=

2 × 2

1

= 4 = 22

Hence,

25

23

= 22 or

25

23

= 25 −^3

This is the second law of indices, which demon-

strates thatwhen dividing two numbers having

the same base, the index in the denominator is

subtracted from the index in the numerator.

(3) ( 35 )^2 = 35 ×^2 = 310 and ( 22 )^3 = 22 ×^3 = 26

This is the third law of indices, which demon-

strates thatwhen a number which is raised to

a power is raised to a further power, the indices

are multiplied.