## What is exponentiation?

Where multiplication is a shorthand notation for the sum of the same term, exponentiation is the same for multiplication: the same term is multiplied many times. The term is called the *base* and how many times it is multiplied is given by the *exponent*. For example, we want to multiply 3 five times. Instead of writing it out, we can use the following notation:

In this example 3 is the base and 5 is the exponent. We read as *“three to the power of five”* (or sometimes *“three to the five”*). The operation can be read as *“raising three to the power of 5″*. We also use the word *degree* as in “three has a degree of five” in this example.

## Special exponents

By definition, every number that is raised to the power of zero is one:

(1)

An exponent of one has no effect: .

When the exponent is 2, we also call this a square. For example we raise a variable to the power of 2: . Because this equation is the same as the area of a square with side , we call this a square. Raising something to the power of two is called squaring it.

Likewise, when the exponent is 3, we call it a cube because is the equation for the volume of a cube with side .

We can also use a variable for the exponent, for example . ( to the power of ) is like multiplying with itself times.

## Negative exponents

When the exponent is negative, we get a fraction where the denominator is the exponential term with a positive exponent. For example consider raising to the power of :

Some examples:

(2)

## Fractions in the exponent

When the exponent is a fraction, the term becomes a root. Let us only consider fractions where the numerator is one (so , etc). When we raise to the power of we get:

For example:

(3)

If you don’t know the meaning yet of : Let’s use from the example above. This means that . In this case because =27. In our other example , we get . In this case the value of because .

Now consider any fraction, not just the ones where the numerator is one. Let us raise to the power of . We get:

The difference is that we need to raise to the power of inside the root. You can also imagine this in two steps. In the first one we raise to the power of and in the second step we raise the result () to the power of (or the other way around). Either way, we get the same result:

We can also have negative fractions. This is a combination of the negative exponent and fraction exponents as discussed so far:

We must be careful when we have a negative fraction when the base is negative and (the denominator) is even: for example when (even), . This is the same as but we can’t have a root for a negative number, so there is no solution.

## Rules for exponentiation

Finally, we want to show you some useful rules for exponentiation.

First, let us look at multiplication and exponentiation when the base is the same.

Next, let us look at division when the base is the same:

We can also have different bases but the same exponents. Let us look at multiplication again with different bases and but the same exponent :

And next, division with different bases and the same exponent:

Finally, we can raise raise an exponent to a degree: