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8 CPP
Chapter 7
Indices
Rather than writing 2  2  2 , we can write such a product as 2 3 . This representation is called the index
notation, where 2 is the base and 3 is the index or exponent.
In general, for any positive integer n, a n = a  a  a  a  .......... .  a
n factors
I-
Index Laws:
If m and n are any two positive integers, then:
1) a m  a n = a m+n
To ……………. powers having the same ………. but different
…..., keep the ……….. and ………… the …………
am
= a m−n , m  n , a  0
n
a
To divide two powers having same ……… but different ……., where the ………
in the ……… is greater than that in the ….…..,
keep the base and subtract ……………….
m
a
1
3) n = n − m , m  n , a  0
a
a
To divide two powers having same base different indices, where
the …… in the numerator is …… than that in the denominator,
keep the ……. and ……………….
2)
4) (a m ) n = a mn
When raising a power to a ……., keep the …… and multiply the
………..
II-
Expansion Laws:
For a positive integer n, we have the following:
1) (ab) n = a n b n
n
an
a
2)   = n , b  0 .
b
b
III-
Zero Index Law:
For all a  0 , a 0 = 1 .
IV-
V-
Odd and Even Indices:
•
n
If, then a  0 , a  0
•
a n  0 if n is even
a  0 , then:  n
a  0 if n is odd
Negative Index Law:
If a is any non-zero number and n is an integer, then a − n =
Example: 2 −3 =
1
23
1
, thus a n and a − n are reciprocals.
an