There is always a simple guide to the laws of indices in Mathematics. This set of principles and mathematical theories follow a certain tradition of calculation which must be followed in order to reach a certain sum total or scientific conclusion. In statistics, economics, and finance, an index is a statistical measure of change in a representative group of individual data points.
These data may be derived from any number of sources, including company performance, prices, productivity, and employment. An index is the small floating number that appears after a number or letter. Indices show how many times a number or letter has been multiplied by itself. Laws of indices provide us with rules for simplifying calculations or expressions involving powers of the same base.
This means that the larger number or letter must be the same. We cannot use laws of indices to evaluate calculations when the bases are different. Here are the laws of indices in Mathematics: simple guide to follow in recognizing and mastering them for both personal and professional use. These laws are used while performing algebraic operations on indices and while solving the algebraic expressions, including it:
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Law 1
If a constant or variable has index as ‘0’, then the result will be equal to one, regardless of any base value.
| a0 = 1 |
Example: 50 = 1, 120 = 1, y0= 1
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Law 2
If the index is a negative value, then it can be shown as the reciprocal of the positive index raised to the same variable.
| a-p = 1/ap |
Example: 5-1 = ⅕, 8-3=1/83
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Law 3
To multiply two variables with the same base, we need to add its powers and raise them to that base.
| ap.aq = ap+q |
Example: 52.53 = 52+3 = 55
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Law 4
To divide two variables with the same base, we need to subtract the power of denominator from the power of numerator and raise it to that base.
| ap/aq = ap-q |
Example: 104/102 = 104-2 = 102
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Law 5
When a variable with some index is again raised with different index, then both the indices are multiplied together raised to the power of the same base.
| (ap)q = apq |
Example: (82)3 = 82.3 = 86
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Law 6
When two variables with different bases, but same indices are multiplied together, we have to multiply its base and raise the same index to multiplied variables.
| ap.bp = (ab)p |
Example: 32.52 = (3 x 5)2 = 152
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Law 7
When two variables with different bases, but same indices are divided, we are required to divide the bases and raise the same index to it.
| ap/bp = (a/b)p |
Example: 32/52 = (⅗)2
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Law 8
An index in the form of a fraction can be represented as the radical form.
| ap/q = q√ap |
Example: 61/2 = √6
Basically, the laws of indices follow these fundamental rules:
- Multiplying indices
- Dividing indicies
- Brackets with indices
- Power of 0
- Negative indices
- Fractional indices
