Download Lesson 2 - Indices

Lesson 2- Laws of Indices
 Objectives
 To know what indices are
 To 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
5
7
7 is the
NUMBER
75 & 24 are numbers in INDEX FORM
Oct 2011
INTO Foundation L2
5 is the
Combining numbers
5x5x5 x2x2x2x2
= 53 x 24
We can not write this any more simply
Can ONLY combine BASE NUMBERS if they are the same
Oct 2011
INTO Foundation L2
Rule 1 : Multiplication
26 x 24
= 210
24 x 22
= 26
35 x 37
= 312
General Rule
Law 1
am x an = am+n
Oct 2011
INTO Foundation L2
Rule 2 : Division
26 ÷ 24
= 22
25 ÷ 22
= 23
35 ÷ 37
= 3-2
General Rule
Law 2
am ÷ an = am-n
Oct 2011
INTO Foundation L2
Rule 3 : Brackets
(26)2
= 26 x 26
(35)3
= 35 x 35 x 35
= 212
= 315
General Rule
Law 3
(am)n = am x n
Oct 2011
INTO Foundation L2
Rule 4 : Index of 0
How could you get an answer of 30 ?
35 ÷ 35
= 35-5
30 = 1
= 30
General Rule
Law 4
a0 = 1
Oct 2011
INTO Foundation L2
Putting them together?
26 x 24
23
= 210
23
= 27
35 x 37
4
3
3
= 312
342
=2
= 38
25 x 2 = 28
24 x 22
26
Oct 2011
INTO Foundation L2
Works with algebra too!
a6 x a4
= a10
b5 x b7
= b12
c5 x c 3
c4
= c8
= c4
c4 -2
a8 = a
a5 x a3 =
a4 x a6
a10
Oct 2011
INTO Foundation L2
..and a mixture…
2a3 x 3a4 = 2 x 3 x a3 x a4 = 6a7
8a6 ÷ 4a4 = (8 ÷ 4) x (a6 ÷ a4) = 2a2
28a6
4a4
Oct 2011
2
INTO Foundation L2
= 2a2
Fractional indices
 (Using Law 1) We could write
1
2
x  x x x
1
But
So
Oct 2011
x
xx
INTO Foundation L2
1
2
x  x
1
2
Fractional Indices
 Similarly
1
3
1
3
x x x x
1
3
x x x x
3
3
3
1
3
So x  3 x
General Rule
Law 5
1
n
a n a
Oct 2011
INTO Foundation L2
Negative Index Numbers.
Simplify the expression below:
5
3
3
5
57
5
7
=5
Write the original expression
again as a quotient:
5 5 5

5 5 5 5 5 5 5
Oct 2011
To understand this result
fully consider the
following:
-4
Expand the numerator and
the denominator:
Cancel out as many fives as possible:
1

5 5 5 5
Write as a power of five:
1
 4
5
Now compare the two results:
INTO Foundation L2
Negative Indices
 The last Index rule
General Rule
Law 6
a-m = 1
am
Oct 2011
INTO Foundation L2
Summary
Rule 1 : Multiplication of Indices.
anxa
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 5 : For fractional indices
a1/n = n√a
Oct 2011
INTO Foundation L2
Rule 3 : For Powers Of
Index Numbers.
a 0 = …..
Exercises
 Section 2- Working with Indices
 Additional Questions if you get that far!
Oct 2011
INTO Foundation L2
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?
Oct 2011
INTO Foundation L2
Calculations involving standard form
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?
Rearrange
speed =
distance
to give
time
time =
8.32 × 107
Time to reach Mars =
2.6 × 103
= 3.2 × 104 hours
This is 8.32 ÷
2.6
Oct 2011
INTO Foundation L2
This is
107 ÷ 103
distance
speed
Calculations involving standard form
Use 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
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
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