Lesson 2- Laws of Indices
Objectives
To know what indices areTo learn the rules of indices
Oct 2011 INTO Foundation L2
What are Indices?
Indices provide a way of writing numbers in a more compact and convenient form
Indices is the plural of Index
An Index is often referred to as a power
Oct 2011 INTO Foundation L2
For example
5 x 5 x 5 = 53
2 x 2 x 2 x 2 = 24
7 x 7 x 7x 7 x 7 = 75
7 is the BASE NUMBER
5 is the INDEX orPOWER
75 & 24 are numbers in INDEX FORM
Oct 2011 INTO Foundation L2
75
Combining numbers
5 x 5 x 5 x 2 x 2 x 2 x 2
= 53 x 24
We can not write this any more simply
Can ONLY combine BASE NUMBERS if they are the same
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Rule 1 : Multiplication
26 x 24 = 210
24 x 22 = 26
35 x 37 = 312
General RuleLaw 1
am x an = am+n
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Rule 2 : Division
26 ÷ 24 = 22
25 ÷ 22 = 23
35 ÷ 37 = 3-2
General RuleLaw 2
am ÷ an = am-n
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Rule 3 : Brackets
(26)2 = 26 x 26 = 212
(35)3 = 35 x 35 x 35 = 315
General RuleLaw 3
(am)n = am x n
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Rule 4 : Index of 0
How could you get an answer of 30 ?
35 ÷ 35 = 35-5 = 30
30 = 1 General RuleLaw 4
a0 = 1
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Putting them together?
25 x 23
24 x 22= 28
26
26 x 24
23= 210
23
35 x 37
34= 312
34
= 27
= 38
= 22
Oct 2011 INTO Foundation L2
Works with algebra too!
a5 x a3
a4 x a6= a8
a10
= a-2
a6 x a4 = a10
b5 x b7 = b12
c5 x c3
c4= c8
c4= c4
Oct 2011 INTO Foundation L2
..and a mixture…
2a3 x 3a4 = 6a7= 2 x 3 x a3 x a4
8a6 ÷ 4a4 = 2a2= (8 ÷ 4) x (a6 ÷ a4)
8a6 4a4
22
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= 2a2
Fractional indices
(Using Law 1) We could write
21
21
1 xxxx
xxx But
So 21
xx
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Fractional Indices
Similarly
331
333
31
31
31
xxSo
xxxx
xxxx
General RuleLaw 5
nn aa 1
Oct 2011 INTO Foundation L2
Negative Index Numbers.Simplify the expression below:
5 3 5 7 = 5 - 4 To understand this result fully consider the following:
Write the original expression again as a quotient:
Expand the numerator and the denominator:
5555555555
7
3
55
Cancel out as many fives as possible:
55551
Write as a power of five:
Now compare the two results:451
Oct 2011 INTO Foundation L2
Negative Indices
The last Index rule
am
General RuleLaw 6
a-m = 1
Oct 2011 INTO Foundation L2
SummaryRule 1 : Multiplication of Indices.
a n x a m =………Rule 2 : Division of Indices.
a n a m = …….Rule 4 : For Powers Of Index Numbers.( a m ) n = …..
Rule 6 : For negative indices
a - m =……. Rule 3 : For Powers Of Index Numbers. a 0 = …..Rule 5 : For fractional indices
a1/n = n√aOct 2011 INTO Foundation L2
Exercises
Section 2- Working with Indices
Additional Questions if you get that far!
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Travelling to Mars
How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
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Calculations involving standard form
Time to reach Mars =8.32 × 107
2.6 × 103
= 3.2 × 104 hours
Rearrange speed =distance
timetime =
distancespeed
to give
This is 8.32 ÷ 2.6
This is 107 ÷ 103
How long would it take a space ship travelling at an average speed of 2.6 × 103 km/h to reach Mars 8.32 × 107 km away?
Oct 2011 INTO Foundation L2
Calculations involving standard formUse your calculator to work out how long 3.2 × 104 hours is
in years.
You can enter 3.2 × 104 into your calculator using the EXP key:
3 . 2 EXP 4
Divide by 24 to give the equivalent number of days.
Divide by 365 to give the equivalent number of years.
3.2 × 104 hours is over 3½ years.
Oct 2011 INTO Foundation L2