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Indices
Chapter 1
2014 – Year 10 Mathematical Methods
Review of Index Laws
Some numbers can be written in mathematical shorthand if the number
is the product of "repeating numbers”.
Example: a7 = a × a × a × a × a × a × a = aaaaaaa
Index and base form
64 = 26
 The 10 is called the index number
 The 2 is called the base number
The plural of “index” is “indices”
Another name for index form is power form or power notation
26 is read as: two to the power of 6
Review of Index Laws
Index Law 1
Index Law 2
Index Law 3
23 x 25 = 23+5
= 28
26
24
23
23
In general terms
In general terms
In general terms
am x an = am+n
am = am-n
an
a0 = 1
=
26-4
=22
= 23-3
=20 = 1
Review of Index Laws
Index Law 4
Index Law 5
(24)2
(2 x 3)4 = 23 x 34
= 24 X 2
= 28
In general terms
In general terms
(am)n = am x n
(a x b)m = am x bm
= amn
Index Law 6
In general terms
Examples
Solve:
Solve:
m2n6p2 x m3np4
6x3y5
2xy2
=
m2+3n6+1p2+4
=
m5n7p6
=3x3-1y5-2
=3x2y3
Examples
Which of the following is equivalent to (x½)6?
A.
x6½
B. x3
Using law 4
We get:
(am)n = am x n
= x½x 6
= amn
C. 6x½
D. ½x6
= x3
B
Examples
Which of the following is equivalent to (2y⅔)3?
8y2
Using law 4
We get:
B. 2y2
(am)n = am x n
= 23y⅔ x 3
A.
= amn
C. 8y3
D. 2y3
= 8y2
A
Negative Indices
Lets have a look at this example of Index Law 2
y2
y4
=y-2
It can also be written as
yxy
yxyxyxy
Therefore we know y-2 also can be written as
Seventh Index Law
a-n = 1n
a
1
y2
or
1
y2
Negative Indices
•
All index laws apply to terms with negative indices
• Always express answers with positive indices unless
otherwise instructed
• Numbers and pronumerals without an index are
understood to have an index of 1 e.g. 2 = 21
Examples
Write the numerical value of:
Express the following with a positive index:
Examples
• Simplify these algebraic expression:
HINT – remove the brackets first, then use the index laws and then
express with positive indices.
Fractional Indices
•
Fractional indices are those which are expressed as
fractions.
Fractional Indices
Fractional Indices
Fractional Indices
Combining Index Laws
When more than one index law is used to simplify an
expression, the following steps can be taken.
Step 1: If an expression contains brackets, expand
them first.
Step 2: If an expression is a fraction, simplify each
numerator and denominator, then divide (simplify
across then down).
Step 3: Express the final answer with positive indices.
Combining Index Laws
Simplify :
Combining Index Laws
Simplify:
Combining Index Laws
Simplify:
Combining Index Laws
Simplify:
Combining Index Laws
Simplify:
Combining Index Laws
Simplify:
Combining Index Laws
• Simplification of expressions with indices often
involves application of more than one Index law.
• If an expression contains brackets, they should be
removed first.
• If the expression contains fractions, simplify across
then down.
• When dividing fractions, change ÷ to × and flip the
second fraction (multiply and flip).
• Express the final answer with positive indices.