# Powers & The Laws Of Indicies

## Powers

To ensure that you properly understand this topic, we recommend you to read the indices introduction. As, you probably already know, indices or powers are numbers that tell us how often a number is to be multiplied by itself in a mathematical expression. They appear quite frequently, in numerical and algebraic expressions. The concept of a power is quite simple but just to ensure that you have understood everything in the introduction and you are ready to proceed, look at the examples underneath. Laws of Indices Now that you are quite familiar with indices, you are ready to move on the the laws of indices. These laws are very important when multiplying and diving powers as well as when using them in algebraic expressions. There are three laws of indices.
LAW 1: The first law of indices tells us that when multiplying two identical numbers together that have different powers (eg: 2² x 2³), the answer will be the same number to the power of both exponents added together. In algebraic form, this rule look like this:                                        . The a represents the number and n and m represent the powers. Here is an example of this rule in action. Example: In the example, we have added 2 and 3 together to give us 5. So, the solution is 3 to the power of 5.
LAW 2:   The second law of indices tells us that when dividing a number with an exponent by the same number with an exponent, we have to subtract the powers. In algebraic form, this rule is as follows                            . The a represents the number that is divided by itself and m and n represent the powers. Here is an example for this rule. Example: As you can see, the powers have been subtracted (5-3=2). So the solution is 4 to the power of 2.
LAW 3: The third and last law tells us that when we have to multiply a power in a bracket, by another power outside the bracket, we have to multiply the two powers together to get the answer. In algebraic form, this rule looks like this                              . The a represents the number in the bracket while the m and n represent the two powers (one inside and one outside of the bracket). Here is an example in which this rule is applied. Example:   In this example, the powers were multiplied together to give the answer which is 3 to the power of 6.

## To the Power of 0

There is another rule with regards to indices that originates from Law 2. This rule states that anything to the power of 0 is equal to 1. In the example, underneath, you will see that x to the power of 0 is always equal to 1. Example: In the expression above, you can see that x to the power of 0 is equal to one. In the expression underneath, is the proof that shows that this is true. Since x³/x³ is x to the power of 0 (according to the second law of indices) and a number divided by itself is always 1 (in this case x³), it must be true that x to the power of 0 is 1. More Complicated Exponents Now that you know almost everything there is to know about simple numerical exponents, you should move on and try to solve some fractional and negative exponents. Or you can give surds a try which involve roots. Please share this page if you like it or found it helpful!
Example 1 In this example, we have 4 to the power of 3 which is equal to 4 x 4 x 4. The answer to the problem is 64.
Example 2 In this example, we are working with a very large power: 6. As a result we have to use 2 six times in the multiplication.
Example 3 In this example, we have to use 5 four times in the multiplication which gives us a very large answer.
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