Indices and Factors
Squares, Cubes, Square Roots and Cube Roots

When a number is multiplied by itself, we say that the number has been squared.

For example, 3 squared means 3 × 3 = 9. This is written as 32 = 9 .
We could also say that 9 is the square of 3.

When a number is cubed it is written down 3 times and multiplied.

For example 2 cubed means 2 × 2 × 2 = 8. This is written as 23 = 8 .
We could also say that 8 is the cube of 2.

Sometimes the reverse process is needed to answer questions such as:

What number squared gives 25?

The answer would be 5. We say that 5 is the square root of 25, or write = 5.
Another question might be:

What number cubed gives 8?

The answer would be 2. We would say that the cube root of 8 is 2.

We could also write = 2 .

Worked Examples

1

Find

(a)
82 Show me 82 = 8 × 8 = 64
(b)
42 Show me 42 = 4 × 4 = 16
(c)
53 Show me 53 = 5 × 5 × 5 = 125

Use your answers to find
(d)
Show me = 8   because   82 = 64
(e)
Show me = 4   because   42 = 16
(f)
Show me = 5   because   53 = 125

Exercises

Find

(a)
52
(b)
62
(c)
12
(d)
72
Use your answers to find
(e)
(f)
(g)
(h)

Find

(a)
33
(b)
43
(c)
63
(d)
103
Use your answers to find
(e)
(f)
(g)
(h)

Find

(a)
102
(b)
22
(c)
42
(d)
72
(e)
82
(f)
92
(g)
13
(h)
73
(i)
83
(j)
02
(k)
03
(l)
23

Find

(a)
(b)
(c)
(d)
(e)
(f)

Use a calculator to find

(a)
122
(b)
112
(c)
153
(d)
133
(e)
132
(f)
152
(g)
202
(h)
113

Without a calculator, find

(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)

Find

(a)
62 + 42
(b)
32 − 22
(c)
102 + 42
(d)
32 + 42
(e)
52 − 32
(f)
43 + 23
(g)
13 + 103
(h)
62 + 82

Information

On average, a human heart beats 75 times a minute, 4 500 times an hour, 108 000 times a day, 39 420 000 times a year and 3 153 600 000 times for someone who lives 80 years.