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# Algebra Expressions with Exponents

## Calculate values of numbers with exponents

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Algebra Expressions with Exponents

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*}

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*}

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

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

### 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*}

### Vocabulary Language: English

Evaluate

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Expanded Form

Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Expression

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Integer

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...
Parentheses

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
substitute

substitute

In algebra, to substitute means to replace a variable or term with a specific value.
Volume

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.