Index Laws

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Indices

25

Base Value

Power (Indice)

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When multiplying terms with the same base we add the powers.

Eg 1: y3 x y4 = y7

Eg 2: 23 x 25 = 28

Eg 3: 3 m2 x 2 m3 = 6 m5

1st Index Law: Multiplication

Eg 4: c2d5 x 6c5d = 6 c7d6

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98 ÷ 95 =

2nd Index Law: Division

9 x 9 x 9 x 9 x 9 x 9 x 9 x 9

9 x 9 x 9 x 9 x 9= 93

So although it has a divide sign, when we are dividing indices we actually subtract them.However this can only be done when there is the same base!

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When dividing terms with the same base we subtract the powers.

eg 1: p 8 ÷ p 2 = 2p

p8p 6

eg 2: 20x7 ÷ 4x3 3

7

4

20

x

x

1

5 5 x4

eg 3: 415

10

m

m52

3 32 m or 3

2m

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You must multiply the powers together, remembering

Multiply everything inside the brackets individually.

(x2)4 = x8

(5b3)2 =25b6

(-3y)2 =9y2

(-2n4)3 =-8n12

3rd Index Law: Power to a further power.

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4th Index Law: Power of zero

Any term to the power of 0, always equals 1.

d0 = 1

50 = 1

40 x 6 = 1 x 6 = 6

132 ÷ 132 = 130 = 1

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h2 x h3 =h5

-3 x 4y =

-12y

2w x 5w2 =

10w3

4b3 x -2b4 =-8b7

(x2)4 =x8

(5b3)2 =25b6

(-3y)2 =9y2

(-2n4)3 =-8n12

Mental Maths:

3

5

a

aa 2

a

a

2

2

2

a

2

5

3

6

x

x2x 3

3

5

15

12

h

h

5

4 2h

5

2

2

6

t

t3

3

t

x

x 4)3( 81x 3

27

32

20

16

nm

nm55

4

m

n

d0 = 1a5

8z0 x 5z5 =

40z5

52 x 5 x 50 =

53

S0 x 42 x s0 =

42

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(3x4)3

= (3x4) x (3x4) x (3x4)= 3 x 3 x 3 x 4 x 4 x 4 = 33 x 43

(Using the first index law)

5th Index Law: Removing Brackets (Products)

You must remember that if there is a letter or number by itself, it is to the power of 1.

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6th Index Law: Removing Brackets (quotient)

x x x x x

=

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Revision 25 = 2 x 2 x 2 x 2 x 2 = 32

Operation Rule Example

Multiplying Add powers y5 x y4 = y9

Dividing Subtract powers

Powers of Powers

Multiply powers

Powers of zero Always = 1 ao = 1

Four rules for working with powers

25powerindex orexponent

base

25

7

aa

a

1243 xx

Same base

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Remember that:

am x an = am+n eg y3 x y 4 = y7

eg

(am)n = amn eg (y 2)3 = y 6

(ab)m = ambm eg (2y)4 = 16y4

nmn

m

aa

a

These rules work the same if the base is a number eg: 23 x 22

=25

5

7

4

442

(32)4 = 38

42

6

yy

y

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7th Index Law: Negative Indices

6-2 =

34 ÷ 39 =

=

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8th Index Law: Fractional Indices

= √36

=

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Important numbers to remember:2 = 21

4 = 22

8 = 23

16 = 24

32 = 25

64 = 26

128 =

27

256 =

28

512 =

29

3 = 31

9 = 32

27 = 33

81 = 34

243 =

35

5 = 51

25 = 52

125 = 53

625 = 54

3125 =

55

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