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This includes the Laws of indices that provide us with rules for simplifying calculations or expressions involving powers of the same base.

1.

2.1

The laws of indices

A power, or an index, is used to write a product of numbers very compactly. The plural of

index is indices. In this leaflet we remind you of how this is done, and state a number of rules,

or laws, which can be used to simplify expressions involving indices.

1. Powers, or indices

We write the expression

3 × 3 × 3 × 3 as 34

We read this as ‘three to the power four’.

z × z × z = z3

We read this as ‘z to the power three’ or ‘z cubed’.

In the expression bc , the index is c and the number b is called the base. Your calculator will

probably have a button to evaluate powers of numbers. It may be marked xy . Check this, and

then use your calculator to verify that

74 = 2401 and 255 = 9765625

1. Without using a calculator work out the value of

2 2 3

1 1 2

a) 42 , b) 53 , c) 25 , d) 2

, e) 3

, f) 5

.

2. Write the following expressions more concisely by using an index.

a a a

a) a × a × a × a, b) (yz) × (yz) × (yz), c) b

× b

× b

.

1. a) 16, b) 125, c) 32, d) 41 , e) 91 , f) 8

125

.

3

a

2. a) a4 , b) (yz)3 , c) b

.

2. The laws of indices

To manipulate expressions involving indices we use rules known as the laws of indices. The

laws should be used precisely as they are stated - do not be tempted to make up variations of

your own! The three most important laws are given here:

www.mathcentre.ac.uk 2.1.1 c Pearson Education Ltd 2000

2.1

The laws of indices

A power, or an index, is used to write a product of numbers very compactly. The plural of

index is indices. In this leaflet we remind you of how this is done, and state a number of rules,

or laws, which can be used to simplify expressions involving indices.

1. Powers, or indices

We write the expression

3 × 3 × 3 × 3 as 34

We read this as ‘three to the power four’.

z × z × z = z3

We read this as ‘z to the power three’ or ‘z cubed’.

In the expression bc , the index is c and the number b is called the base. Your calculator will

probably have a button to evaluate powers of numbers. It may be marked xy . Check this, and

then use your calculator to verify that

74 = 2401 and 255 = 9765625

1. Without using a calculator work out the value of

2 2 3

1 1 2

a) 42 , b) 53 , c) 25 , d) 2

, e) 3

, f) 5

.

2. Write the following expressions more concisely by using an index.

a a a

a) a × a × a × a, b) (yz) × (yz) × (yz), c) b

× b

× b

.

1. a) 16, b) 125, c) 32, d) 41 , e) 91 , f) 8

125

.

3

a

2. a) a4 , b) (yz)3 , c) b

.

2. The laws of indices

To manipulate expressions involving indices we use rules known as the laws of indices. The

laws should be used precisely as they are stated - do not be tempted to make up variations of

your own! The three most important laws are given here:

www.mathcentre.ac.uk 2.1.1 c Pearson Education Ltd 2000

2.
First law

am × an = am+n

When expressions with the same base are multiplied, the indices are added.

We can write

76 × 74 = 76+4 = 710

You could verify this by evaluating both sides separately.

z 4 × z 3 = z 4+3 = z 7

Second Law

am

= am−n

an

When expressions with the same base are divided, the indices are subtracted.

We can write

85 z7

= 85−3 = 82 and similarly = z 7−4 = z 3

8 3 z 4

Third law

(am )n = amn

Note that m and n have been multiplied to yield the new index mn.

(64 )2 = 64×2 = 68 and (ex )y = exy

It will also be useful to note the following important results:

a0 = 1, a1 = a

1. In each case choose an appropriate law to simplify the expression:

y7 x8

a) 53 × 513 , b) 813 ÷ 85 , c) x6 × x5 , d) (a3 )4 , e) y3

, f) x7

.

2. Use one of the laws to simplify, if possible, a6 × b5 .

1. a) 516 , b) 88 , c) x11 , d) a12 , e) y 4 , f) x1 = x.

2. This cannot be simplified because the bases are not the same.

www.mathcentre.ac.uk 2.1.2 c Pearson Education Ltd 2000

am × an = am+n

When expressions with the same base are multiplied, the indices are added.

We can write

76 × 74 = 76+4 = 710

You could verify this by evaluating both sides separately.

z 4 × z 3 = z 4+3 = z 7

Second Law

am

= am−n

an

When expressions with the same base are divided, the indices are subtracted.

We can write

85 z7

= 85−3 = 82 and similarly = z 7−4 = z 3

8 3 z 4

Third law

(am )n = amn

Note that m and n have been multiplied to yield the new index mn.

(64 )2 = 64×2 = 68 and (ex )y = exy

It will also be useful to note the following important results:

a0 = 1, a1 = a

1. In each case choose an appropriate law to simplify the expression:

y7 x8

a) 53 × 513 , b) 813 ÷ 85 , c) x6 × x5 , d) (a3 )4 , e) y3

, f) x7

.

2. Use one of the laws to simplify, if possible, a6 × b5 .

1. a) 516 , b) 88 , c) x11 , d) a12 , e) y 4 , f) x1 = x.

2. This cannot be simplified because the bases are not the same.

www.mathcentre.ac.uk 2.1.2 c Pearson Education Ltd 2000