## Powers (Indices) and Roots

### Introduction

 52 is shorthand for 5 x 5 53 is shorthand for 5 x 5 x 5 54 is shorthand for 5 x 5 x 5 x 5 etc., etc.

The usual way of describing expressions like this is : "5 to the power 3",   "5 to the power 4", and so on.

The first two in the list have special 'names' which could also be used as well :

 52 :   five to the power two or      five squared 53 :   five to the power three or       five cubed

### Square Numbers   (Squares, for short)

Proceeding by example :- the following are square numbers

 9 because 9 = 3 X 3 therefore 32 (three squared) = 9 16 because 16 = 4 X 4 therefore 42 (four squared) = 16 25 because 25 = 5 X 5 therefore 52 (five squared) = 25 144 because 144 = 12 X 12 therefore 122 (twelve squared) = 144

To illustrate the use of the name square : a square with sides of length 4 cms., will have an area of

4 X 4 = 16 cm2 (16 square centimeters)

Quick Quiz     What is the square of the following numbers

 1.    7 2.    6 3.    12 4.    13 5.    20 6.    100 7.   41 8.    19 9.    30

### Cube Numbers (Cubes,  for short)

Taking the above ideas one step further :- the following are cube numbers

 8 because 8 = 2 X 2 X 2 therefore 23 (two cubed) = 8 27 because 27 = 3 X 3 X 3 therefore 33 (three cubed) = 27 64 because 64 = 4 X 4 X 4 therefore 43 (four cubed) = 64 1728 because 1728 = 12 X 12 X 12 therefore 123 (twelve cubed) = 1728

To illustrate the use of the name cube : a cube with sides of length 4 cms., will have an area of

4 X 4 X 4 = 64 cm3 (64 cubic centimeters)

Quick Quiz     What is the cube of the following numbers

 1.    7 2.   6 3.   12 4.   13 5.   20 6.   100 7.   41 8.   19 9.   30

### Powers of Ten

102 = 10 x 10 = 100

103 = 10 x 10 x 10 = 1 000

104 = 10 x 10 x 10 x 10 = 10 000

105 = 10 x 10 x 10 x 10 x 10 = 100 000

106 = 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000

Note that the power of ten, as shown on the left, is equal to the number of zeroes in the expressions on the right.

### Square Roots

#### Proceed by example $\sqrt{9} = 3$ because 3 x 3 = 9

This is using the inverse procedure to that described in the section on square numbers. You can appreciate that not all integers have integer square roots - in fact, integer square roots are the exception.

The following numbers all have integer square roots : 25, 36, 144.

The following do not have integer square roots : 2, 7, 11.

For example, the square root of 2 is 1.414, to 3 decimal places. You will have to use your calculator for these more complicated roots.

But do not use your calculator for the square roots of the lower integers - these should be memorized.

You should remember from your previous work that two minus numbers multiplied together produce a positive number, e.g.

(-3) × (-3) = 9

so you can see that actually -3 is also a square root of 9.

In fact, every number has two square roots. The 'additional' one is just the negative of the other.

Quick Quiz     What is the square root of the following numbers

 1.     121 2.    169 3.    49 4.    3 5.    70