What Is An Index Number.You should know that:
8 x 8 x 8 x 8 x 8 x 8 = 8 6 We say“eight to the power of 6”.
The power of 6 is an index number.
The plural of index numbers is indices.
What are the indices in the expressions below:
(a) 3 x 5 4 (b) 36 9 + 34 (c) 3 x 2
4 9 3 & 2
The number eight is the base number.
If the index number is 1 we just write the base number Eg 81=8
Multiplication Of Indices.We know that : 7 x 7x 7 x 7 x 7 x 7 x 7 x 7 = 7 8
But we can also simplify expressions such as :
6 3 x 6 4 To simplify:
(1) Expand the expression.= (6 x 6 x 6) x (6 x 6 x 6 x 6)
(2) How many 6’s do you now have?
7
(3) Now write the expression as a single power of 6.
= 6 7
Key Result.
6 3 x 6 4 = 6 7
We can also simplify expressions such as :
2 x 3 To simplify:
(1) Expand the expression.= ( x ) x ( x x )
(2) How many ’s do you now have?
5
(3) Now write the expression as a single power of .
= 5
Key Result.
2 x 3= 5
Multiplication Of Indices.Multiplication Of Indices.
Using the previous example try to simplify the following expressions:
(1) 3 7 x 3 4
= 3 11
(2) 5 x 9
= 14
(3) p11 x p 7 x p 8
= p 26
We can now write down our first rule of index numbers:
First Index Law: Multiplication of Indices.
a n x a m = a n + m
NB: This rule only applies to indices with a common base number. We cannot simplify 11 x p 7 as and p are different bases.
What Goes In The Box ? 1Simplify the expressions below :
(1) 6 4 x 6 3
(2) g 7 x g 2
(3) d6 x d
(4) 14 9 x 14 12
(5) 25 x 30
(6) 2 2 x 2 3 x 2 5
(7) p 7 x p 10 x p
(8) 5 20 x 5 30 x 5 50
= 6 7
= g 9
= d 7
= 14 21
= 55
= 2 10
= p 18
= 5 100
Division Of Indices.Consider the expression:
47 88 The expression can be written as a quotient:
4
7
8
8 Now expand the numerator
and denominator.
8888
8888888
How many eights will cancel from the top and the bottom ?
4
Cancel and simplify.
888 =8 3
Result:
8 7 8 4= 8 3
Using the previous result simplify the expressions below:
(1) 3 9 3 2
= 3 7
(2) 11 6
= 5
(3) p 24 p 13
= p 11
Second Index Law: Division of Indices.
a n a m = a n - m
We can now write down our second rule of index numbers:
What Goes In The Box ? 2Simplify the expressions below :
(1) 5 9 5 2
(2) p 12 p 5
(3) 19 6 19
(4) 15 10
(5) b 40 b 20
(6) 2 32 2 27
(7) h 70 h 39
(8) 5 200 5 180
=5 7
= p 7
= 19 5
= 5
= b 20
= 2 5
= h 31
= 5 20
Zero IndexConsider the expression:
33 22
3
3
2
2
222
222
8
8
=1 Since the two results should be the same:
2 3 2 3= 20 = 1
Using 2nd Index Law
33 22
3
3
2
2
332 02
Zero IndexUsing the previous result simplify the expressions below:
(1) 30 (2) 6 6
= 0
(3) (p 24 )0
= 1
Third Index Law : Zero Index
a 0 = 1 (where a 0)
We can now write down our third rule of index numbers:
= 1
= 1
Powers Of Indices.Consider the expression below:
( 2 3 ) 2
To appreciate this expression fully do the following:
Expand the term inside the bracket.
= ( 2 x 2 x 2 ) 2Square the contents of the bracket.
= ( 2 x 2 x 2 ) x (2 x 2 x 2 ) Now write the expression as a power of 2.
= 2 6
Result: ( 2 3 ) 2 = 2 6
Use the result on the previous slide to simplify the following expressions:
(1) ( 4 2 ) 4 (2) ( 7 5 ) 4 (3) ( 8 7 ) 6
= 4 8 = 7 20 = 8 42
We can now write down our fourth rule of index numbers:
Fourth Index Law: For Powers Of Index Numbers.
( a m ) n = a m x n
Fifth and Sixth Index Laws
These are really variations of the Fourth Index Law
Fifth Index Law:
(a x b)m = am x bm
Sixth Index Law: m
b
a
mb
ma
Fifth and Sixth Index Laws - Variations
Fifth Index Law:
(2a x 3b)2 = 22a2 x 32b2
= 4a2 x 9b2
= 36a2b2
Sixth Index Law: 2
3
2
b
a
22
22
3
2
b
a
Do not forget to raise the constants to the power as well?Eg:
2
2
9
4
b
a
PracticeMaths Quest 10 Exercice 1A (page 5-6)
Questions 1, 2, 3, 4 & 6: a, b, h, iQuestion 7: a to f
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:
5555555
555
7
3
5
5
Cancel out as many fives as possible:
5555
1
Write as a power of five:
Now compare the two results:45
1
The result on the previous slide allows us to see the following results:
Turn the following powers into fractions:
(1) 32
32
1
8
1
(2) 43
43
1
81
1
(3) 610
610
1
1000000
1
We can now write down our seventh rule of index numbers:
For Negative Indices:.
a - mma
1
More On Negative Indices.Simplify the expressions below leaving your answer as a positive index number each time:
(1)5
96
3
33
)5(963 5963
83
(2)28
34
77
77
)2(8
34
7
7
6
1
7
7
617 77
77
1
What Goes In The Box ? 3Change the expressions below to fractions:
Simplify the expressions below leaving your answer with a positive index number at all times:
(1)52 (2)
33
3
2
2
4
(3) (4)
3
2
3
6
3
65
4
44
(5) 1110
67
77
77
(6) (7)246
342
333
333
32
1
27
1
2
1
4
3
44 27 33
1
What Goes In The Box ? 4Simplify the expressions below leaving your answer as a positive index number.
(1) 54 )7(63 )5(
(2) (3)37 )10(
(4)342 )88( (5) 523 )77( (6) 1056 )1111(
207 185
1 2110
188 57 11011
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