1. Index Laws
Miss Hudson’s Maths

2. Indices 25
Power (Indice)
Base Value
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3. 1st Index Law: Multiplication
When multiplying terms with the same base we add the powers.
Eg 1: y3 x y4 = y7
Eg 2: x = 28
Eg 3: 3 m2 x 2 m3 = 6 m5
Eg 4: c2d5 x 6c5d =
6 c7d6
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4. 2nd Index Law: Division 98 ÷ 95 = 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9 = 93
= 93
9 x 9 x 9 x 9 x 9
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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5. When dividing terms with the same base we subtract the powers.
eg 1: p 8 ÷ p 2 =
p 6 5 5 x4
eg 2: 20x7 ÷ 4x3 1 2 m
eg 3: or 3
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6. 3rd Index Law: Power to a further power.
You must multiply the powers together, remembering
Multiply everything inside the brackets individually.
(x2)4 = x8
(-3y)2 =
9y2
(5b3)2 =
25b6
(-2n4)3 =
-8n12
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7. 4th Index Law: Power of zero
Any term to the power of 0, always equals 1.
d0 = 1
40 x 6 = 1 x 6 = 6
50 = 1
132 ÷ 132 = 130 = 1
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8. Mental Maths: h2 x h3 = h5 d0 = 1 -3 x 4y = -12y 2x 3 2w x 5w2 = 10w3
8z0 x 5z5 =
40z5
4b3 x -2b4 =
-8b7
52 x 5 x 50 = 53
(x2)4 = x8
S0 x 42 x s0 = 42
(5b3)2 =
25b6
(-3y)2 =
9y2
(-2n4)3 =
-8n12
81x 3
a 2
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9. 5th Index Law: Removing Brackets (Products)
(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)
You must remember that if there is a letter or number by itself, it is to the power of 1.
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10. 6th Index Law: Removing Brackets (quotient)=
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11. Four rules for working with powers
Revision
Miss Hudson’s Maths 25
power
index or
exponent
base
25 = 2 x 2 x 2 x 2 x 2 = 32
Four rules for working with powers
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
Same base

12. These rules work the same if the base is a number eg:
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
These rules work the same
if the base is a number eg:
23 x 22 = 25 42
(32)4 = 38
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13. 7th Index Law: Negative Indices
6-2 =
34 ÷ 39 = =
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14. 8th Index Law: Fractional Indices
= √36 =
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15. Important numbers to remember:
2 = 21
4 = 22
8 = 23
16 = 24
32 = 25
64 = 26
128 = 27
256 = 28
512 = 29
5 = 51
25 = 52
125 = 53
625 = 54
3125 = 55
3 = 31
9 = 32
27 = 33
81 = 34
243 = 35
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