PROPERTIES OF POWERS

Now it's time for us to learn more about powers. As many algebraic operations, powers have their own properties, which are easily understood once we remember what are the exponent and the base of a power. There are five major properties of powers, which serve as a pathway to factoring, as well as to many other areas of mathematics, including geometry.

Property 1

During multiplication of powers with the same base, the exponents are added while the base remains unchanged.

Let us review some examples. Indeed, in accordance with the definition of power with natural exponent:

In general form:

Important: pay attention to the fact that base does not change. See examples below:

Property 2

During division of powers with the same bases, the base remains the same while the exponents are subtracted.

Or, since division is a particular case of a fraction and vice-versa:

Which is still the same operation of division, after all. Yet again, let us review some examples:

Also look at these examples:

Property 3

During powering of the power, the base remains the same, while exponents are multiplied by each-other.

See for yourself:

In general form this property will look like this:

See examples below:

Property 4

This property will be different. Before it we were dealing only with the same bases, this time it is going to change: we are going to deal with the product of different bases. It's enough to remember that many numbers could be represented as products: 12 = 3x4 or 2x6, or 25 = 5x5 and thus:

During powering of the product, each multiplier of this product should be raised to the same power.

There was a reason for us mentioning representing numbers as products: often you will encounter a situation when you have to represent a number as a product. See for yourself: 12³ is just that 12 cubed, but there will be situations when you will have to represent this number as a product. In this case Property 4 will come in handy. 12, of course, could be represented as a product in two ways: 4^.3 and 6^.2 so now we can power both of these products:

This property is so simple that requires no additional explanations, and, at a close inspection, bears an uncanny resemblance to the distributive property of multiplication. In the case of the powers we "distribute" not the factor but the exponent:

or other way around (of course, only is the different bases will have the same exponent: )

Property 5

The power of a fraction can be represented as a fraction of the powers with the numerator and the denominator of this fraction as the bases:

Here, we "distribute" our exponent to both the numerator and the denominator and this is the whole trick. See examples below:

So, now we know 5 fundamental properties of powers. At first this looks simple but later on more complex exercises will teach you how to apply these properties separately, consecutively, and later, in a complex and sophisticated sequences. 

Now let's try a practical assignment..

Practice >

Graphs of linear functions of the form y = mx + b [click]
Properties of powers [click]
Did you know? [click]
The principle of energy conversion [click]
The meaning of an increment [click]
The hydraulic press [click]

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