Remember the tent dilemma from the Identify Whole Number Powers, Exponents and Bases Concept?

Well, the hikers were given a specific tent with specific dimensions. They were given a Kelty Trail Dome 6 tent.

What if a different tent was used? What if many different tents were used?

The square footage of the floor would always have an exponent of 2, but a variable would be needed for the base because different size tents would be being used.

Here is how we could write this.

In this case, a is the length of one side of a square tent.

What if a tent with 8 feet on one side was being used?

What if a tent with 15 feet on one side was being used?

What would the square footage of each tent be?

**This Concept will teach you how to evaluate powers with variable bases. Pay attention and you will know how to work through this at the end of the Concept.**

### Guidance

When we are dealing with numbers, it is often easier to just simplify. It makes more sense to deal with 16 than with

**Yes, and we can simplify by using exponential form and we can also write out the variable expression by using expanded form.**

Write the following in expanded form:

**To write this out, we simply write out each x five times.**

We can work the other way to by taking an variable expression in expanded form and write it in exponential form.

**Our answer is**

**What about when we multiply two variable terms with exponents?**

To do this, we are going to need to follow a few rules.

**The first thing to notice is that these terms have the same base. Both bases are m’s. Because of this, we can simplify the expression quite easily.**

**Let’s write it out in expanded form.**

**Here we have five m’s being multiplied our answer is**

Here is the rule.

Let’s apply this rule to the next one.

**The bases are the same, so we add the exponents.**

**This is the answer.**

**We can also have an exponential term raised to a power. When this happens, one exponent is outside the parentheses. This means something different.**

**Let’s think about what this means. It means that we are multiplying x squared by itself three times. We can write this out in expanded form.**

**Now we are multiplying three bases that are the same so we use Rule 1 and add the exponents.**

**Our answer is**

**We could have multiplied the two exponents in the beginning.**

Here is Rule 2.

Simplify

**Our answer is**

**Anything to the power of 0 equals 1.**

Now it's time for you to try a few on your own.

#### Example A

Write the following in exponential form:

**Solution: a7**

#### Example B

Simplify:

**Solution: a11**

#### Example C

Simplify:

**Solution: x8**

Remember the tent dilemma from the beginning of the Concept? Well let's take a look at it again.

The hikers were given a specific tent with specific dimensions. Remember, they were given a Kelty Trail Dome 6.

What if a different tent was used? What if many different tents were used?

The square footage of the floor would always have an exponent of 2, but a variable would be needed for the base because different size tents would be being used.

Here is how we could write this.

In this case, a is the length of one side of a square tent.

What if a tent with 8 feet on one side was being used?

What if a tent with 15 feet on one side was being used?

What would the square footage of each tent be?

Here is our solution.

### Vocabulary

- Exponent
- a little number that tells you how many times to multiply the base by itself.

- Base
- the big number in a variable expression with an exponent.

- Exponential Notation
- writing long multiplication using a base and an exponent.

- Expanded Form
- taking a base and an exponent and writing it out as a long multiplication problem.

### Guided Practice

Here is one for you to try on your own.

Simplify:

**Answer**

When we multiply variables with exponents, we add the exponents.

**Our answer is x8.**

### Video Review

- This is a James Sousa video on evaluating powers with variable bases.

### Practice

Directions: Evaluate each expression.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Directions: Simplify the following variable expressions.

14.

15.

16.

17.

18. \begin{align*}(a^5 )(a^2)\end{align*}

19. \begin{align*}(x^9 )(x^3)\end{align*}

20. \begin{align*}(y^4 )(y^5)\end{align*}

Directions: Simplify.

21. \begin{align*}(x^2 )^4\end{align*}

22. \begin{align*}(y^5 )^3\end{align*}

23. \begin{align*}(a^5 )^4\end{align*}

24. \begin{align*}(x^2 )^8\end{align*}

25. \begin{align*}(b^3 )^4\end{align*}