Chapter 7
Powers, roots and laws of
indices
7.1 Introduction
The manipulation of powers and roots is a crucial underlying skill needed in algebra. In this chapter, powers and
roots of numbers are explained, together with the laws
of indices.
Many worked examples are included to help understanding.
7.2 Powers and roots
7.2.1 Indices
The number 16 is the same as 2 × 2 × 2 × 2, and 2 × 2 ×
2 × 2 can be abbreviated to 2 4 . When written as 2 4 , 2 is
called the base and the 4 is called the index or power.
2 4 is read as ‘two to the power of four’.
Similarly, 3 5 is read as ‘three to the power of 5’.
When the indices are 2 and 3 they are given special
names; i.e. 2 is called ‘squared’ and 3 is called ‘cubed’.
Thus,
4 2 is called ‘four squared’ rather than ‘4 to the power
of 2’ and
5 3 is called ‘five cubed’ rather than ‘5 to the power of 3’
When no index is shown, the power is 1. For example,
2 means 2 1 .
Problem 1. Evaluate (a) 2 6 (b) 3 4
(a) 2 6 means 2 × 2 × 2 × 2 × 2 × 2 (i.e. 2 multiplied
by itself 6 times), and 2 × 2 × 2 × 2 × 2 × 2 = 64
i.e.
2 6 = 64
(b) 3 4 means 3 × 3 × 3 × 3 (i.e. 3 multiplied by itself
4 times), and 3 × 3 × 3 × 3 = 81
i.e.
3 4 = 81
Problem 2. Change the following to index form:
(a) 32 (b) 625
(a) (i) To express 32 in its lowest factors, 32 is initially
divided by the lowest prime number, i.e. 2.
(ii) 32 ÷ 2 = 16, hence 32 = 2 × 16.
(iii) 16 is also divisible by 2, i.e. 16 = 2 × 8. Thus,
32 = 2 × 2 × 8.
(iv) 8 is also divisible by 2, i.e. 8 = 2 × 4. Thus,
32 = 2 × 2 × 2 × 4.
(v) 4 is also divisible by 2, i.e. 4 = 2 × 2. Thus,
32 = 2 × 2 × 2 × 2 × 2.
(vi) Thus, 32 = 2 5 .
(b) (i) 625 is not divisible by the lowest prime number, i.e. 2. The next prime number is 3 and 625
is not divisible by 3 either. The next prime
number is 5.
(ii) 625 ÷ 5 = 125, i.e. 625 = 5 × 125.
(iii) 125 is also divisible by 5, i.e. 125 = 5 × 25.
Thus, 625 = 5 × 5 × 25.
(iv) 25 is also divisible by 5, i.e. 25 = 5 × 5. Thus,
625 = 5 × 5 × 5 × 5.
(v) Thus, 625 = 5 4 .
DOI: 10.1016/B978-1-85617-697-2.00007-7
Powers, roots and laws of
indices
7.1 Introduction
The manipulation of powers and roots is a crucial underlying skill needed in algebra. In this chapter, powers and
roots of numbers are explained, together with the laws
of indices.
Many worked examples are included to help understanding.
7.2 Powers and roots
7.2.1 Indices
The number 16 is the same as 2 × 2 × 2 × 2, and 2 × 2 ×
2 × 2 can be abbreviated to 2 4 . When written as 2 4 , 2 is
called the base and the 4 is called the index or power.
2 4 is read as ‘two to the power of four’.
Similarly, 3 5 is read as ‘three to the power of 5’.
When the indices are 2 and 3 they are given special
names; i.e. 2 is called ‘squared’ and 3 is called ‘cubed’.
Thus,
4 2 is called ‘four squared’ rather than ‘4 to the power
of 2’ and
5 3 is called ‘five cubed’ rather than ‘5 to the power of 3’
When no index is shown, the power is 1. For example,
2 means 2 1 .
Problem 1. Evaluate (a) 2 6 (b) 3 4
(a) 2 6 means 2 × 2 × 2 × 2 × 2 × 2 (i.e. 2 multiplied
by itself 6 times), and 2 × 2 × 2 × 2 × 2 × 2 = 64
i.e.
2 6 = 64
(b) 3 4 means 3 × 3 × 3 × 3 (i.e. 3 multiplied by itself
4 times), and 3 × 3 × 3 × 3 = 81
i.e.
3 4 = 81
Problem 2. Change the following to index form:
(a) 32 (b) 625
(a) (i) To express 32 in its lowest factors, 32 is initially
divided by the lowest prime number, i.e. 2.
(ii) 32 ÷ 2 = 16, hence 32 = 2 × 16.
(iii) 16 is also divisible by 2, i.e. 16 = 2 × 8. Thus,
32 = 2 × 2 × 8.
(iv) 8 is also divisible by 2, i.e. 8 = 2 × 4. Thus,
32 = 2 × 2 × 2 × 4.
(v) 4 is also divisible by 2, i.e. 4 = 2 × 2. Thus,
32 = 2 × 2 × 2 × 2 × 2.
(vi) Thus, 32 = 2 5 .
(b) (i) 625 is not divisible by the lowest prime number, i.e. 2. The next prime number is 3 and 625
is not divisible by 3 either. The next prime
number is 5.
(ii) 625 ÷ 5 = 125, i.e. 625 = 5 × 125.
(iii) 125 is also divisible by 5, i.e. 125 = 5 × 25.
Thus, 625 = 5 × 5 × 25.
(iv) 25 is also divisible by 5, i.e. 25 = 5 × 5. Thus,
625 = 5 × 5 × 5 × 5.
(v) Thus, 625 = 5 4 .
DOI: 10.1016/B978-1-85617-697-2.00007-7