Remember the tent dilemma from the Whole Number Exponents?

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.

\begin{align*}a^2\end{align*}

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 \begin{align*}4^2\end{align*}. Exponential notation really comes in handy when we’re dealing with variables. It is easier to write \begin{align*}y^{12}\end{align*} than it is to write \begin{align*}yyyyyyyyyyyy\end{align*}.

**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: \begin{align*}x^5\end{align*}

**To write this out, we simply write out each \begin{align*}x\end{align*} five times.**

\begin{align*}x^5=xxxxx\end{align*}

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

\begin{align*}aaaa\end{align*}

**Our answer is** \begin{align*}a^4\end{align*}.

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

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

\begin{align*}(m^3)(m^2)\end{align*}

**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.**

\begin{align*}mmm(mm)\end{align*}

**Here we have five \begin{align*}m\end{align*}’s being multiplied our answer is** \begin{align*}m^5\end{align*}.

Here is the rule.

Let’s apply this rule to the next one.

\begin{align*}(x^6)(x^3)\end{align*}

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

\begin{align*}x^{6+3}= x^9\end{align*}

**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.**

\begin{align*}(x^2)^3\end{align*}

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

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

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

**Our answer is** \begin{align*}x^6\end{align*}.

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

\begin{align*}(x^2)^3= x^{2(3)} =x^6\end{align*}

Here is Rule 2.

Simplify \begin{align*}x^0\end{align*}

**Our answer is** \begin{align*}x^0 = 1\end{align*}

**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: **\begin{align*}aaaaaaa\end{align*}**

**Solution: \begin{align*}a^7\end{align*}**

#### Example B

Simplify: \begin{align*}(a^3)(a^8)\end{align*}

**Solution: \begin{align*}a^{11}\end{align*}**

#### Example C

Simplify: \begin{align*}(x^4)^2\end{align*}

**Solution: \begin{align*}x^8\end{align*}**

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.

\begin{align*}a^2\end{align*}

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.

\begin{align*}8^2 = 64\end{align*} square feet is the first tent.

\begin{align*}15^2 = 225\end{align*} square feet is the second tent.

### 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:\begin{align*}(x^6)(x^2)\end{align*}

**Answer**

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

**Our answer is \begin{align*}x^8\end{align*}.**

### Video Review

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

### Practice

Directions: Evaluate each expression.

1. \begin{align*}2^3\end{align*}

2. \begin{align*}4^2\end{align*}

3. \begin{align*}5^2\end{align*}

4. \begin{align*}9^0\end{align*}

5. \begin{align*}5^3\end{align*}

6. \begin{align*}2^6\end{align*}

7. \begin{align*}3^3\end{align*}

8. \begin{align*}3^2+4^2\end{align*}

9. \begin{align*}5^3+2^2\end{align*}

10. \begin{align*}6^2+2^3\end{align*}

11. \begin{align*}6^2-5^2\end{align*}

12. \begin{align*}2^4-2^2\end{align*}

13. \begin{align*}7^2+3^3+2^2\end{align*}

Directions: Simplify the following variable expressions.

14. \begin{align*}(m^2)(m^5)\end{align*}

15. \begin{align*}(x^3)(x^4)\end{align*}

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

17. \begin{align*}(b^7 )(b^2)\end{align*}

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*}