Incredible Indices

53 24 xx 8x8

When we multiply, we the powersADD

ababb 23 52 826 baWhen we divide, we the powersSUBTRACT

46 212 aa 26a

cab

cba3

52

7

14 22abNote the c terms cancel out completely because they have the same powers!

87

32

12

4

cab

cba743 cb

a Remember to place the letters in the top or bottom depending on where the biggest power was !

83

22

10

5

ycx

ycx62

1

xc

Remember after you do all cancelling, and there’s nothing left in the TOP, you MUST put a 1.

xy

yx

5

15 42

1

3 3xy 33xyRemember after you do all cancelling, and there’s nothing left in the BOTTOM, you DON’T have to put a 1.

Anything raised to power zero equals…

150 = 1

a0 = 1

( – 356)0 = 1

(2ab2c8)0 = 1

b0 + 7 = 8

3 – t0 = 2

4a0 = 4 x0

= 4 1 = 4

a0 + 3b0 + 5c0 = 9

Raising powers to powers using brackets

32 )(a means222 aaa which is

6a

The quick way to get the answer is to

the powers!!MULTIPLY

25 )(b 10b

25 )3( b 109b

37 )3( y 2127 y

325 )6( ba 615216 ba

Combined operations 1

32 53 4

57 4

32

3 4

xyx y

xy x y

=6 8 3 15

2 2 7 20

4 27

9 4

x y x y

x y x y

6 8 3 15

2 2 7 20

4 27

9 4

x y x y

x y x y

3

=9 23

9 22

3x y

x y

= 3y

=

Combined operations 225

2 3

2 4

3( ) 3

xy x

xy y

5 2

3 6 2

2 9

3 16

xy y

x y x

3

87

5 6

3

8

xy

x y=

=4

3

8

y

x

Remember with divisions you need to turn the 2nd fraction upside down!

This is called multiplying by the reciprocal.

Negative Powers

a – 3 = 3

1

a

b – 7 = 7

1

b

(3b) – 2 =

2

1

(3 )b 2

1

9b

3b – 2 = 3 b –

2 = 3

2

1

b 2

3

b

a – 3b 4 c – 7 d

4

3 7

b d

a c Note – negative power terms go to

the bottom. Others to the top!!

1a

b

1b

a

When a fraction is raised to a negative power, you can invert the fraction and change the power to a positive!2

2

3

b

ay

23

2

ay

b

2 2

2

9

4

a y

b

33 5

4 2

2

5

a b

a b

34 2

3 5

5

2

a b

a b

inverting the stuff in the brackets and changing the outside power from – 3 to 3

12 6

9 15

125

8

a b

a b

getting rid of the brackets. 53 is 125, and 23 is 8. Powers are all multiplied by 3

12 9

15 6

125

8

a a

b b Moving anything with a negative power to

the opposite part of the fraction.

21

21

125

8

a

b

More “exotic” power problems!

164

281

33

x

xx

412

333

2)2(

2)2(

x

xxRecognising all the big numbers as powers of 2

422

393

22

22

x

xxRemoving the brackets. Remember when you remove brackets, you multiply powers

x

x

26

62

2

2

Multiplying along top & bottom. Remember when you multiply, you add powers

Now we’ll divide the top by the bottom and this means we subtract the powers. (2x – 6) – (6 – 2x) = 4x – 12

= 24x – 12 (ans)

1.

2.31

33

1812

46

xx

xx

3212

323

)32()32(

)2()32(

xx

xx

Express your answer as powers of 2 and 3

Breaking all 4 big numbers down into their prime factors

623122

2633

3232

232

xxxx

xxx

Removing the brackets. Remember when you remove brackets, you multiply powers

51

33

32

32

xx

xxSelectively multiplying terms with the same big numbers & adding powers

Now divide the top by the bottom and this means we subtract the

powers (top minus bottom)

2’s: (-x + 3) – ( – 1 – x) = 4 3’s: (x – 3) – ( x – 5 ) = 2

= 24 × 32 Or… 144

Simplify

Fractional powers

aa 2

1

525252

1

416162

1

3992

1

aa 2

1

33

1

aa

46464 33

1

32727 33

1

288 33

1

10

1

100

1

100

1100

2

12

1

n mnm

aa

Example 3 232

88 3 64 4

mnnm

aa OR

OR

2332

88 2)2( 4

Using left formula

Using right formula

You can use either of these formulas, but the one on the right usually avoids getting really big

numbers and so makes the whole process more manageable

Work out 82/3 using the graphics

Examples using fractional powers

2

1106 )9( ba 533 ba=

Note that 91/2 = 9 which is 3.

The rest of the powers have been multiplied

6 x ½ = 3 and 10 x ½ = 5

326

84

)(25

16

yx

yx

First, get rid of any brackets and switch terms with negative powers to the other part of the fraction

8618

4

25

16

yyx

x

Note the y – 8 has been switched from top to bottom

439

2

5

4

yyx

x

79

2

5

4

yx

x 775

4

yx