Exponents Lessons

Before we dive into simplifying exponents, let’s take some time to learn exactly what an exponent is. An exponent is a superscript,
or small number written at the top right corner of a number, variable, or set of parentheses. An example of one is shown


This tells you to multiply 1 by the number as many times as the exponent says.
The example above is 2 raised to the third power (raised to the third power
means the exponent is 3). This is equivalent to the multiplication problem below, because there is a 1 multiplied by 2 three

1 * 2 * 2 * 2

As you can see, the 1 * 2 * 2 * 2 can be simplified to 8 which is the answer to the problem.


Examine the next problem:


This problem is changed to the multiplication problem below. Because of the order of operations (explained in a later lesson)
the exponent is simplified first and then, the negative sign is added to the answer.

1 * 3 * 3 * 3 * 3 * 3 * 3

As you can see, the multiplication simplifies to the number -729. You can do the work for this in your head, on the margin
of your paper, or using a calculator if allowed.


The next problem is shown below:


This time the -3 is inside parentheses. Instead of carrying down the negative sign, each 3 is made negative.

1 * -3 * -3 * -3 * -3 * -3 * -3

As you can see, the multiplication simplifies to 729. Note that aside from
the result of the previous example being negative, the result
here is the same. Pay careful attention to finding exponents when negative
signs are involved, as this is a common source of error.

Special Cases

The zero exponent

Each one of these problems is solved by using a 1 multiplied by the number,
the amount of times the exponent indicates. If the exponent is 0, the 1 isn’t
multiplied by the number at all. Therefore, the answer is 1.
This is an important rule to remember.



Zero with an exponent:

In most cases zero with an exponent can be calculated like any other number
and exponent.


1 * 0 * 0 * 0 * 0

Note that as long as 1 is being multiplied by at least one 0, the end result
is 0. Therefore we can conclude that 0 to any positive exponent is always zero.

Another special case occurs with 00. Zero with an exponent of
zero is undefined, and cannot be calculated.

Be careful not to the rules for zero exponents! Zero to any positive power
is always zero, because no matter how many times you multiply the 1 by
zero the answer will always be zero. But 00 is undefined.

The 1 exponent:

Consider this example in which rasies a number is raised to the first power.


1 * 51

If you try any similar example such as 101 or 1001, you
will find that the result is always the original number or base. This is
because 1 times any other number is always equal to the second number.

So to simplify the case where a number is raised to the first power, we can
simply remove the exponent.


Ten to Any Power

This tip can save you a lot of time: Ten to any power is simply the number 1
followed by a number of zeros indicated by the exponent.
An example is shown below.



Note that the result is a one followed by five zeros because the exponent on
10 was 5.

Reading Exponents

In general, a problem such as


is read as “Five to the tenth power.”

Special Cases for Reading Exponents

Certain exponents have special ways of being pronounced. These make it a
little easier to say, but it is not necessary that you use them.

  • The Second Power: 32 – may be read as either “Three to the second power,” or “Three squared.”
  • The Third Power: 103 – may be read as either “Ten to the third power,” or “10 cubed.”

Exponents of Numbers Resources

Practice Problems / Worksheet
Practice all of the methods you learned in this lesson here.

Next Lesson:
Exponents of Variables

Learn how to simplify a variable inside a parenthesis with an exponent.


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