incredible indices. 8x88x8 when we multiply, wethe powers add when we divide, wethe powers subtract...
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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