Chapter 2 Power and Index
In p^{n}, p is called the base and n is called the power or index
In (5)^{7}, 5 is te base and 7 is the power
In , is the base and 7 is the power
And
=
Remarks: In the exponential notation, the base can be any rational number and the power can be any integer.
Note: If the power of a rational number is 1 its value will be the rational number itself
i.e. (10)^{1} = 10, and in general
.
Example: Find the value of
Solution:
Product law for exponents: If p is a nonzero rational number and m and n are two positive integers then p^{m} x p^{n} = p ^{m+n}
also p^{m} x p^{n} x p^{r} x p^{s} = p^{m+n+r+s}
also (p^{m})^{n} = p^{mn}
Quotient law for exponents: It p is a non zero rational number and m and n are two positive integers
then p^{m} ÷ p^{n} = p^{mn} for m > n
and p^{m} ÷ p^{n} = for m < n
If power is zero (o): If p is a nonzero rational number then p^{o} = 1
If power is (1): If p is a nonzero rational number then p^{1} denotes the reciprocal of p and (p)^{1} =
A negative integer as power
p^{m} =
other laws of exponents
p^{m} x q^{m} = (p x q)^{m}
Few examples showing the application of laws of exponents.
Example1: Simplify (a)
Solution: a)
= 2^{2} x 3^{4+2} = 2^{2} x 3^{2}
Example2: Find m if
Solution
(a) LHS=
RHS =
Equating LHS and RHS
Because base is same, powers must be equal
So 2m + 1 = 27
or 2m = 27 – 1
= 28
or m = 14
(b) LHS =
RHS = 2^{m}
So 2^{m} = 2^{5}
Or m = 5.
Example: Solve for x
a) 3^{x} = 81 b) (7^{2x})^{2} = (2401)^{1}
Solution a) RHS = 81 = 3^{4}
So 3^{x} = 3^{4} or x = 4
Solution b) RHS= (2401)^{1} = (7^{4})^{1} = 7^{4}
LHS = (7^{2x})^{2} = 7^{4x}
Equating LHS and RHS
7^{4X} = 7^{4}
Base in same, powers must be equal
4x = 4
Or x = 1
What is the difference between exponents
= p^{m} x p^{m} x p^{m} x p^{m}  n times
=p^{m+m+m+m}  n times
=p^{m n}
Where as =
Let us simplify it with the help of an example
Find the difference between
(2^{2})^{3} = 2^{2} x 2^{2} x 2^{2} = 2^{6}
= 2^{2x2x2 }= 2^{8}
So the diff. is very clear.
56 videos104 docs95 tests

1. What is power in mathematics? 
2. How do you calculate the value of a power? 
3. What is an index or exponent in mathematics? 
4. How does the concept of power relate to reallife situations? 
5. What are some common mistakes to avoid when working with powers and exponents? 
56 videos104 docs95 tests


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