algebra_indicessurds_na_notes_dl.pdf

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Indices and SurdsRevision Notes

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For a positive integer n, na is defined as:

10 a , where 0a e.g. 150

n

n

a

a 1 e.g.

3

3

2

12

nn aa 1

, where a > 0 and n is a positive integer e.g.33

1

55

n mn

m

aa , where a > 0 and both m and n are positive integers

e.g.3 23

2

55

Indices with a Common Base

1. nmnm aaa e.g. 624 222

2. nmn

m

aa

a , where 0a e.g. 22

4

22

2

3. mnnm aa e.g. 824 22

Indices with a Common Index but Different Bases

Laws of Indices

Fractional Indices

Zero and Negative Indices

Indices

where a is called the base, and n, the index or exponent.

n factors

,... aaaaan


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1. mmm baba )( e.g. 222 )24(24

2.

m

m

m

b

a

b

a

, where 0b e.g.

2

2

2

2

4

2

4

Other Laws

nn

a

b

b

a

e.g.

22

3

4

4

3

n

m

n

m

a

b

b

a

e.g.5

2

5

2

3

4

4

3

An equation with a variable occurring in the index or exponent is known asexponential equation.

ba x is the simplest form of an exponential equation

If b can be expressed as na ,

ba x n xaa n x then .

A surd has a general form of n a , where a can be any natural number.

A surd is an irrational number which cannot be expressed as a fraction in the form

n

m

. Hence, it is a root, which cannot be evaluated exactly

For natural numbers a and b,

abba e.g. 632

Laws of Surds

Surds

Exponential Equations


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b

a

b

aba e.g.

5

353

aaa e.g. 666

anmanam )( e.g. 757372

anmanam )( e.g. 112114116

22 bnambnambnam e.g.

62

5423

5423

54235423

22

22

a is in its simplest form if a does not contain a perfect square factor.

If bma 2 such that ,0m 0b and a = bm2 , then its simplest form is bm .

bnam and bnam are conjugate surds. The product of a pair of conjugate surds is a rational number

To rationalise the denominator of a surd means to make the denominator a rationalnumber.

Rationalising Simple Denominator

b

ba

b

b

b

a

b

a e.g.

3

32

3

3

3

2

3

2

Rationalising the Denominator

Conjugate Surds

Sim lification of a where a > 0


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bc

ca

c

c

cb

a

cb

a e.g.

12

32

3

3

34

2

34

2

Rationalising a Compound Denominator

To rationalise a compound denominator we use the concept of conjugate surds.

cb

cba

cb

cba

cb

cb

cb

a

cb

a

22

e.g.

2

352

35

352

35

35

35

2

35

2

cb

cba

cb

cba

cb

cb

cb

a

cb

a

22

e.g.

3

254

25

254

25

25

25

4

25

4

For some equations, the following result for equality of surds may be used:

.andIf qn pmaq panm

Solving Equations Involving Surds

algebra_indicessurds_na_notes_dl.pdf

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