Laws of indices: Index Notation is said to be a method of reducing/shortening the product of numbers that have equal factors or numbers that are the same.
For instance if we have 4 x 4 x 4 x 4. We can shorten the product of the numbers by taking one of the equal factors “4” and raising it to the power of the number of its occurrence (that is the number of times the factors appear). “4” appears 4 times, which means that 4 x 4 x 4 x 4 = 4⁴.
In general if we have n x n x n x n x n = n raised to power 5 = n^5, “n” can be any number. The “n” is known as the BASE while 5 is called the power or index.
Note that INDICES is the plural of INDEX
Laws of Indices: There are different laws governing index Notation which are:
1). Multiplication law:
When you multiply numbers with equal bases, their powers are being added.
For example: 7⁴ x 7²
But if a group of different bases are multiplied, the different numbers are collected in their “like” factors as one group to give the product.
E.g 2² x 3³ x 2³ x 5⁴ x 5² x 3
= 2² x 2³ x 3³ x 3 x 5⁴ x 5²
2). Division law:
When two numbers that have equal factors/bases are being divided, this very law is telling us to take one of the factors and subtract the power of the divisor from the power of the dividend.
3). Power law:
This law is telling us that if a number/base which is raised to a particular power is again raised to another power, it is said to be in powers and the two powers are made to multiply each other for the final result.
4). Product power law:
Product power law is saying that when product of different factors are being raised to a certain power maybe in a bracket, the power is distributed over each and every of the factor in that same bracket.
5). Zero Index:
This law states that when any number, quantity or anything at all is raised to a power zero, the result is 1(one). a° = 1, b° = 1, 1° = 1, 2° = 1, 7° = 1. If any number at all is raised to power zero, the answer is 1 except zero raised to power zero (0°) which is undefined. This law can be proven even using the Division law.
6). Negative Index:
If a number is raised to a negative Index, the result is the reciprocal of that number with index turning positive. This can also be proven by using the Division law.
7). Roots power:
Roots power is the method of showing the indices in different roots by its powers such as square root, cube root and so on.
Any number “x” raised to power 1/2 is the square root of that number √x, any number “x” raised to power 1/3 = cube root of that number 3√x, any number raised to power 1/4 is the fourth root of that number 4√x and so on.
8). Fractional Index:
Some times a number is raised to a power which is a fraction, be it proper or improper fraction other than half, one-third, one-fourth etc. The approach in which we resolve this is using the denominator as the root to find the nth root of the number and later raised the result to the power of the numerator.