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 equation 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)   equation

An exponent of one has no effect: equation.

When the exponent is 2, we also call this a square. For example we raise a variable equation to the power of 2: equation. Because this equation is the same as the area of a square with side equation, 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 equation is the equation for the volume of a cube with side equation.

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


Some examples:

(2)   equation

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 equation, equation etc). When we raise equation to the power of equation we get:


For example:

(3)   equation

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

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


The difference is that we need to raise equation to the power of equation inside the root. You can also imagine this in two steps. In the first one we raise equation to the power of equation and in the second step we raise the result (equation) to the power of equation (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 equation (the denominator) is even: for example when equation (even), equation. This is the same as equation 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 equation and equation but the same exponent equation:


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


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