In this post we are going to explain the properties of powers. Furthermore, we will try to justify the reason why they are fulfilled. But first, let’s remember what a power is.

##### What is a power?

A power is the product of a number by itself a certain number of times. In other words, **a power is a multiplication of equal factors** . This idea can be written in the following way:

The repeating factor, **a** , is known as the * base* , while the number of times it is repeated,

**n**, is called the

*.*

**exponent**You can find more information about what powers are in this other post. Once we have refreshed what a power is, let’s continue with its properties. But first, let’s establish that the base has to be different from 0 … After all, who is interested in multiplying by zero?

##### Product of powers of the same base

Sometimes we find products of powers that share the same base. For example:

2 ^{3} × 2 ^{5}

6 ^{2} × 6 ^{7}

In those cases we may be interested in using this property. Note that the following equality is true because multiplication is associative and the order in which the products are made does not matter.

This property can be read as * “the product of powers of the same base is equal to maintaining the base and adding the exponents”* .

Note that it can also be read from right to left, that is, that the factors of one power can be grouped as desired to give rise to the product of two others with minor exponents. For example:

2 ^{10} = 2 ^{7} × 2 ^{3}

##### Ratio of powers of the same base

Similar to what happens with multiplication, it also happens with division. Remember that in a fraction you can simplify the factors repeated in the numerator and denominator. Notice!

This property is stated as * “the quotient of powers of the same base is equal to maintaining the base and subtracting the exponents”* . As was the case before, this equality can be read in the other direction and done, for example:

5 ^{3} = 5 ^{7 – 4} = 5 ^{7} : 5 ^{4}

##### Powers with null exponent

Many textbooks, when explaining the properties of powers, treat this situation as a separate case. In my opinion it is not necessary because, if we look at the immediately previous property, a power with null exponent can be understood as a quotient of equal powers. For example, 2 ^{3} : 2 ^{3} . But also 5 ^{2} : 5 ^{2} . Y 3465 ^{7} : 3465 ^{7}

In general, any non-zero base with any exponent. And how much is a fraction worth with the numerator and denominator equal? Well 1, of course.

Therefore, it is said that * “any number raised to 0 is 1”* .

##### Negative exponent powers

In a more or less similar way it happens with the powers of negative exponent. If we look at the power quotient property of the same base, if **m** is greater than **n **, the subtraction **n – m** is a negative number. Therefore, it is not surprising that a power with a negative exponent is a fraction with factors only in the denominator.

In other words, *“a power with a negative exponent is equal to a unit fraction that has the same power in the denominator but with the positive exponent*