Our final operator has the highest precedence, is binary, and is usually invisible. Here, we are going to discuss the raise to a power operation. Basically, a raise to a power operation looks like this:

2^{3}

The ** 2** is called the

This power, 2^{3}, evaluates to
eight because 2^{3} means two times itself three
times, that is, two times two times two.

There is no ** visible** operator. It is the

This raise to a power operation has precedence over all the binary operators (multiplication, division, addition, subtraction) and unary operators (positive and negative signs). For example, consider this example:

4 * 2^{3}

We have two operations present here:
** multiplication** and

32 = 4 * 2^{3}

Here is a harder one to understand:

^{-}4^{2}

Now, the ** negative sign** out front must wait
till the

^{-}16 = ^{-}4^{2}

In other words, ^{-}4^{2}
does not mean negative four times negative four. That would
be positive sixteen. Some calculators give this result; so,
be careful and make sure that you understand how the
calculator that you are using works.

The exponent may be negative. Consider this expression:

4^{-3}

The ** negative sign** on the

1 / 4^{3} = 4^{-3}

Although the above notation is not incorrect in any way, perhaps this is more clear:

The exponent can be a fraction. Taking a root, such as a square root or a cube root, is actually the raising of a number to a fractional power. Here are some examples:

Now, if the exponent contains some arithmetic, all of that arithmetic must be done before you can clearly see with what power you are working. In other words, the exponent itself can be an expression with operators and operands. For example, consider this:

3^{2 + 4}

In the above example the exponent is the
expression '2 + 4', which evaluates to six. So the entire
expression, 3^{2 + 4}, evaluates to the sixth power
of three, or equivalently three times three times three times
three times three times three, or seven hundred and twenty
nine. So:

3^{2 + 4} = 3^{6} = 729

Here is an interesting situation:

What does that mean? Well, it means 2 raised to some power. But what power? Well, the exponent for 2 contains some arithmetic which itself contains a raise to the power operation. The exponent for two is the fourth power of three, or eighty-one. In notation that looks this way:

By the way, the eighty-first power of two is quite a large number. My calculator reads: 2,417,851,639,229,258,349,412,352. I would suspect that is correct, but I really have no common experience to check it against.

Sometimes an operator is shown for the
raise to the power operation. This is regularity true when
such an expression must be entered into a device that does
not permit superscripts, such as the graphics calculator like
EZ Graph. Under such conditions a
** caret**, or

3^4

Stands for this:

3^{4}

Where are you?

Other Representations Besides Numbers for Values

Introduction to Operators and Operands

More about Operators and Operands

*Here: Raise to a Power Operation*

And Now Including Variables and Functions

Just a Few Notes about Multiplication