The index of a number is also known as the power or exponent. It actually tells us how many times we have to multiply the number by itself. For example, we consider
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Definition of Index
The power/exponent raised to a number is called the index of that number.
Mathematically, we understand the index of a number as follows. For a real number
Note that the plural form of index is indices. ♣
Laws of Indices
We now discuss the laws of indices (or the rules of indices). This will help us to solve the problems of indices.
• Zero index rule:
Thus, if the index of any non-zero number is
• Negative index rule:
Thus, if the index of any number is negative, then the value will be the reciprocal of the positive index raised to that number itself. For example,
• Quotient rule of indices:
(i)
Examples:
(i)
• Product rule of indices:
(i)
Examples:
(i)
• Fraction rule of indices:
So if the index is a fraction, then the value can be expressed as a radical form. For example,
Some Remarks on Indices
(R1) Simple proof of
Proof:
(R2) Meaning of
(R3) Meaning of
Before providing solved examples, we now summarize the fundamental laws of indices.
Solved Examples of Indices
Ex 1: Simplify
Solution.
Note that
∴
Ex 2: Calculate
Solution.
Ex 3: If
Solution.
Case 1:
Now
So
⇒
Case 2:
Note that
Conclusion:
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