Exponents of Variables Lesson

The last lesson explained how to simplify exponents of numbers by multiplying as shown below. You know that 3 squared is the same as 1 * 3 * 3.

Exponents of variables work the same way – the exponent indicates how many times 1 is multiplied by the base of the exponent. Take a look at the example below.

The first problem we will work on is below. It doesn’t contain a variable, but it will help us to learn how to simplify a similar problem with a varable in place of the first 3.

Normally, you would simplify this problem by simplifying the inside of the parentheses first:

Then, simplify the exponent outside the parentheses.

This method gives a correct answer, but there is an easier way.

Exponents of Variables

We will be solving the same problem again:

This time, instead simplifying inside of the parentheses first, we will “distribute” the exponent of the parentheses to the inside of the parentheses.

Now the only thing left to do is simplify the exponent that is left.

As you can see this method also gives an answer of 729.

Exponents of Variables

The first example with variables is

We will try simplifying it the first way, by simplifying the inside of the parentheses followed by simplifying the exponent on the outside.

Now that the inside is simplified, the exponent on the parentheses indicates that the expression is equivalent to a 1 multiplied by the parentheses, three times. As you can see x is being multiplied 6 times, hence the answer x to the sixth power.

Exponents of Variables

Again, the problem we are working is

As with the second number example earlier in this lesson, simply multiply the two exponents:

Then remove the parentheses, and as you can see the answer is the same.

Exponents of Variables

The problem below has two key differences.

  • First, it has a term with two variables, and as you can see the exponent from outside the parentheses must multiply EACH of them.
  • Second, there is a negative sign inside the parentheses. Since the exponent on the parentheses is 3, the negative sign is written in front of the term three times. Then the multiple signs are simplified.

Both the problem above and below this have a negative sign inside a set of parentheses which is raised to some power. If you did a lot of these you’d notice that when the parentheses are raised to an odd power such as 3, the answer will be negative. If
the parentheses are raised to an even power like the one below, the answer will be positive.

The last problem, shown below has a negative sign outisde the parentheses. Again, because of the Order of Operations which is presented in a later lesson, the exponent must be simplified before you do anything with the negative sign. Look at the work

Note that even though the exponent on the parentheses was a 4 which is an even number, the final answer is negative. This is because the negative sign was outside of the parentheses, not inside as in the previous example.

Exponents of Variables Resources

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

Next Lesson:
Exponents of Polynomials (Parentheses)

Learn how to simplify an exponent of a polynomial, or two or more terms inside a parenthesis.


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