Laws of Indices and their calculations with examples

Indices are another form of important expressions in mathematics. Since there can be several places and incident to simplify mathematical expressions, everyone should know the laws of indices and how to apply those laws in the correct way.
This article provides all the laws of indices that would be very useful in solving mathematical equations especially for ordinary level school/college students and of course for everybody who needs simplifications.

Here we go!

All the basic laws in indices can be defined as follows with examples.

1. When there are multiple multiplications applied for numbers with different powers but the same base, then all the powers can be added into the same, single based number. Figure 1 shows examples as below.

   

   Figure 1
2. When there are multiple divisions applied for numbers with different powers but the same base, then the powers can be subtracted accordingly in the same order as they appeared and get only one power with the same, single based number. Figure 2 shows examples as below.
   
   

   

   Figure 2

   
   

   As it shows in example 1 and 3 in figure 2, there are basic rules in indices simplification. Everyone should know them to apply in the correct way and get the most simplified answer using them. Therefore, the rules are as follows.

   • The 0^th power of any number is 1. It can be represented as,
     

   •  Any power of the base 0 is 0. It can be represented as,

   
   

   

   
   

   Continuing the laws from 3 onward . . .
   
   
3. Any negative power of any base can be made positive by exchange the position of the base between the numerator and the denominator. It can be shown in figure 3 below.
   

   

   Figure 3

   
   
4.  When there are extending powers for a single base, then those powers get multiplied into one power.
   It can be shown in figure 4 below.

   

   Figure 4

   
   
5. If there are numbers inside the radical sign, the symbol can be replaced with an index accordingly. It helps to simplify expressions or solving equations with indices. It can be shown in figure 5 below.

   

   Figure 5

These laws can be used anywhere to simplify expressions or solving equations. The next important fact with indices is the solving equations with indices. There are basically two facts to keep in memory to solve such equations. Those are pretty simple and straight forward as follows.

1. The equation should be simplified until having two numbers in both sided of the equal sign.
2. Then, if the bases are identical (except for 0, 1, and negatives) in both sides, powers MUST BE equal (identical). On the other hand, if the powers are identical (except for 0, 1, and negatives) in both sides, bases MUST BE equal (identical).

IMPORTANT- The second fact above has some restrictions when it applies in expressions. It is because of the behavior of the value zero and one (0 & 1). Thus the complete rule with all conditions can be shown according to Figure 6 and some examples are shown there as below.

Figure 6

Up to now, very important laws and rules of indices are discussed in this article. They can be used anywhere in the correct way as explained to simplify mathematical expressions or to solve equations. Figure 7 provides some more examples of equations in indices and simplification below for further awareness. 

Figure 7

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