IndicesChapter 1

2014 – Year 10 Mathematical Methods

Review of Index LawsSome numbers can be written in mathematical shorthand if the number is the product of "repeating numbers”.

Example: a7 = a × a × a × a × a × a × a = aaaaaaa

Index and base form64 = 26

The 10 is called the index number The 2 is called the base number

The plural of “index” is “indices”Another name for index form is power form or power notation

26 is read as: two to the power of 6

Review of Index Laws

Index Law 2

26

24

In general terms

am

an

Index Law 1

23 x 25 = 23+5

= 28

In general terms

am x an = am+n

Index Law 3

23

23

In general terms

a0 = 1

= 26-4 =22

= am-n

= 23-3 =20 = 1

Review of Index Laws

Index Law 4

(24)2 = 24 X 2 = 28

In general terms

(am)n = am x n

= amn

Index Law 5

(2 x 3)4 = 23 x 34

In general terms

(a x b)m = am x bm

Index Law 6

In general terms

ExamplesSolve:

m2n6p2 x m3np4

= m2+3n6+1p2+4

= m5n7p6

Solve:

6x3y5

2xy2

=3x3-1y5-2

=3x2y3

ExamplesWhich of the following is equivalent to (x½)6?

A. x6½

B. x3

C. 6x½

D. ½x6

Using law 4

(am)n = am x n

= amn

We get:

= x½x 6

= x3

B

ExamplesWhich of the following is equivalent to (2y⅔)3?

A. 8y2

B. 2y2

C. 8y3

D. 2y3

Using law 4

(am)n = am x n

= amn

We get:

= 23y⅔ x 3

= 8y2

A

Negative IndicesLets have a look at this example of Index Law 2

y2 y x y 1 y4 y x y x y x y y2

Therefore we know y-2 also can be written as

Seventh Index Law

a-n =

It can also be written as or=y-2

1 y2

1an

Negative Indices• All index laws apply to terms with negative

indices

• Always express answers with positive indices unless otherwise instructed

• Numbers and pronumerals without an index are understood to have an index of 1 e.g. 2 = 21

ExamplesWrite the numerical value of:

Express the following with a positive index:

Examples• Simplify these algebraic expression: HINT – remove the brackets first, then use the index laws and

then express with positive indices.

Fractional Indices• Fractional indices are those which are expressed

as fractions.

Fractional Indices

Fractional Indices

Fractional Indices

Combining Index LawsWhen more than one index law is used to simplify an expression, the following steps can be taken.

Step 1: If an expression contains brackets, expand them first.

Step 2: If an expression is a fraction, simplify each numerator and denominator, then divide (simplify across then down).

Step 3: Express the final answer with positive indices.

Combining Index LawsSimplify :

Combining Index LawsSimplify:

Combining Index LawsSimplify:

Combining Index LawsSimplify:

Combining Index LawsSimplify:

Combining Index LawsSimplify:

Combining Index Laws• Simplification of expressions with indices often

involves application of more than one Index law.• If an expression contains brackets, they should be

removed first.• If the expression contains fractions, simplify

across then down.• When dividing fractions, change ÷ to × and flip

the second fraction (multiply and flip).• Express the final answer with positive indices.