Whenever we add a number to itself multiple times, we call it addition, but when it is multiplication, then it is called exponentiation and the product is called power of the number. So, a mathematical operation where two numbers are necessary, one is written in the superscript of the other and is called index or exponent and the other number is called base of the number to which the power is raised using the index.
Definition of Index
For any real number \(\ a \) and natural number \(\ n \), \(\ a^n \) is defined as
(i) when \(\ a \) is a positive integer:
\(\ a^n = a×a×a×a×a×....\) up to \(n\) factor
Here, \(\ n \) is called the base of the number and \(\ n \) is called the index or exponent and \(\ a^n \) is called the power of the number.
For example, \(\ 5^3 = 5×5×5 \)
(ii) when \(\ n \) is a negative integer:
\(\ a^{-n} = \frac{1}{a^n}; a ≠ 0 \)
Here, \(\ a^{-n} \) is called the inverse of \(\ a^n \)
Laws of Indices
1. If \(\ a \) is some real number and \(\ m \) and \(\ n \) are some natural numbers,then \(\ {a^m}×{a^n} = a^{m+n} \)
Proof:
LHS = \(\ {a^m}×{a^n} \)
\(\ = a×a×a×a×....\) up to \(m\) factor
\(\ ×a×a×a×....\) up to \(n\) factor
\(\ = a×a×aa×a....\) up to \((m+n)\) factor
\(\ = a^{m+n} \) = RHS
2. If \(\ a \) is real number and \(\ m \) and \(\ n \) are some natural numbers,
then \(\ {a^m}÷{a^n} = a^{m-n} \)
Proof:
Case I: when \(\ m > n \) and \(a ≠ 0\)
LHS = \(\ {a^m}÷{a^n} \)
\(\ = \frac{a^m}{a^n} \)
\(\ = a^m×a^{-n} \)
\(\ = [a×a×a×....\) up to \(m\) factor]
\(\ ×[a×a×a×....\) up to \(-n\) factor]
\(\ = a×a×a×a×.....\) up to \([m+(-n)]\) factor
\(\ = a×a×a×a×a×.....\) up to \((m-n)\) factor
\(\ = a^{m-n} \) = RHS
Case II: when \(\ m = n \)
LHS = \(\ \frac{a^m}{a^n} \)
\(\ = a^m×a^{-n} \)
\(\ = [a×a×a×....\) up to \(m\) factor]
\(\ ×[a×a×a×....\) up to \(-n\) factor]
\(\ = a×a×a×a×.....\) up to \([m+(-n)]\) factor
\(\ = a×a×a×a×a×.....\) up to \((m-n)\) factor
\(\ = a^{m-n} \) = RHS
\(\ ⇨ {a^m}÷{a^n} = a^{m-n}\) ...(1)
Since, \(\ m = n \)
Eqn (1) ⇨ \(\ \frac{a^m}{a^m} = a^{m-m} \) ⇨ \(\ 1 = a^{0} \)
And the result \(\ a^{0} = 1 \) is another law indices, where \(\ a ≠ 0 \)
Case III: when \(\ m < n \)
LHS = \(\ {a^m}/{a^n} \)
\(\ = {1\over{a^na^{-m}}} \)
\(={1\over{[a×a×..~up~to~n~factor]×[a×a×.... up~to~-m~ factor]}}\)
\(\ = {1\over{a×a×a×a×...~up~to~{n+(-m)}~factor}} \)
\(\ = {1\over{a^{n-m}}} \)
\(\ = a^{-(n-m)} \)
\(\ = a^{m-n} \) = RHS
3. If \(\ a \) is a natural number and \(\ m \) and \(\ n \) are some natural numbers, then \(\ {(a^m)}^n = a^{mn} \)
Proof:
LHS = \(\ {(a^m)}^n \)
\(\ = [{a×a×a×a....~ up~ to~ m~ factor}]^n \)
\(\ = [a×a×a×a....~ up~ to~ m~ factor] \)
\(\ ×[a×a×a×a....~ up~ to~ m~ factor] \)
\(\ ×[a×a×a×a....~ up~ to~ m~ factor] \)
\(\ .....~ up~ to~ n~ factor \)
\(\ = a×a×a×a....~ up~ to~ m×n~ factor \)
\(\ = a^{mn} \) = RHS
4. If \(\ a \) and \(\ b \) are two real numbers and \(\ n \) is some natural number, then \(\ \left(ab\right)^n = a^nb^n \)
Proof:
LHS = \(\ \left(ab\right)^n \)
\(\ = (ab)^n \)
\(\ = (ab)(ab)(ab)(ab).....~ n~ factor \)
\(\ = (a×a×a×....~ n~ factor) \)
\(\ (b×b×b×....~ n~ factor) \)
\(\ = a^nb^n \) = RHS
5. If \(\ a \) and \(\ b \) are two real numbers and \(\ n \) is some natural numbers, then \(\ \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \)
Proof:
LHS = \(\ \left(\frac{a}{b}\right)^n \)
\(\ = \left(\frac{a}{b}\right)\left(\frac{a}{b}\right)\left(\frac{a}{b}\right)\left(\frac{a}{b}\right)......up~ to~ n~ factor \)
\(\ = \frac{a×a×a×.... n~ factor}{b×b×b×.... n~ factor} \)
\(\ = \frac{a^n}{b^n} \) = RHS
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