Page 51 - SM inner class 7.cdr

6 -21 7 -21 3 -21 4 - ö 9 5 - ö 9 4 - ö 9
æ
æ
æ
æ
a æ ö
æ
6
6
(d) a ¸ b = ç ÷ (e) ç ö ÷ ¸ ç ö æ ö (f) ç ÷ ¸ ç ÷ = ç ÷
÷
÷ = ç
b è ø è 22 ø è 22 ø è 22 ø è 13 ø è 13 ø è 13 ø
2
C. Power Law 10 = 10 10
´
1
n
We have studied that a m ´ a = a m + n . Let us 10 = 10
0
4 2
use this law to simplify an expression (3 ) . 10 = 1
)
(3 4 2 = 3 4 ´ 3 4 10 - 1 = 1
10
8
= 3 4 + 4 = 3 is the same as 3 4 ´ 2 1 1 1 1
10 - 2 = ´ = =
We solve another expression using the same law. 10 10 10 10 10 2
´
é æ - ö 1 7 ù 2 æ - ö 1 7 æ - ö 1 7 10 -m = 1 = 1
ç ê ÷ ú = ç ÷ ´ ç ÷ 10 10 ´ .... ´ 10 m - 1) m
(
´
ê è 2 ø ú è 2 ø è 2 ø 10
û
ë
times multiplication
7 + 7 14
æ - ö 1 æ - ö 1
= ç ÷ = ç ÷ In general, it can be written as ;
è 2 ø è 2 ø 1
a - m =
7 ´ 2 m
æ - ö 1 a
is also the same as ç ÷
è 2 ø
We can also deduce this law from
+
n
Thus, from the above examples, we can a m ´ a = a m n Suppose n = - m, then we will
deduce that the base remains the same with get,
a new exponent equal to the product of the two a m ´ a - m = a m - m
exponents, that is Þ a m ´ a - m = a 0
) =
(a m n a m ´ n = a mn m - m
Þ a ´ a = a
Zero Exponent Q a = 1
0
By the quotient law, we know that anything m
Divided by a on both sides.
divided by itself is 1 as shown below. m - m
a ´ a 1 - m 1
3 2 3 ´ 3 Þ = Þ a =
= = 1 a m a m a m
3 2 3 ´ 3
Thus, we have another law:
0
This can also be written as 3 2 - 2 = 3 = 1
Any non-zero number raised to any negative
Similarly, power is equal to its reciprocal raised to the
´
´
´
(-2 ) 4 (-2 ) (-2 ) (-2 ) (-2 ) opposite positive power.
= =1
´
´
´
(-2 ) 4 (-2 ) (-2 ) (-2 ) (-2 ) - m 1
i e. a =
.
a m
This can also be written as
0
(-2 ) 4 - 4 = (-2 ) =1 if p - m is a non-zero rational number, then
q
Thus, we can define this law as:
according to the above given law, we have :
Any non-zero rational number with zero
- m m m
exponent is equal to 1. Suppose “a” be any æ pö 1 q æ q ö
ç ÷ = = = ç ÷
non-zero rational number with exponent “0”, è q ø p m p m è pø
then q m
0
a = 1 - m m
æ pö æ q ö
Negative Exponents Thus, ç ÷ = ç ÷
pø
è
q ø
è
Look at the pattern given below.
51
Mathematics In Focus - 7