Casey missed three days of school this week because of a nasty cold. When he returned to school on Thursday he asked his math teacher what he had missed. Miss Brown wrote a problem on a sheet of paper and handed it to Casey. When he arrived home from school Casey looked at his homework which was the following problem:

Casey didn’t know what to do with

In this concept, you will learn how to evaluate numerical expressions involving powers.

### Evaluating Expressions Involving Powers

A **numerical expression** is an expression made up only of numbers and operations but an expression written with a variable in it, is called a **variable expression**.

Both a numerical expression and a variable expression can include powers.

A **power** is the result of multiplying a number by itself one or more times. The number 16 is the fourth power of 2.

**base** and the number expressing the power is called the **exponent**. The exponent tells you how many times to multiply the **base** times itself.

Let’s look at an example.

Evaluate the following expression:

First, expand the powers to see the operations you need to perform.

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right, multiply:

Then, multiply:

Next, from left to right, add:

Then, add:

The answer is 266.

Let’s look at another example.

First, expand the powers to see the operations you need to perform.

Next, write the expression in expanded form.

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.

Then, in order from left to right multiply:

Then, multiply:

Next, from left to right, add:

Then, add:

and write the new expression.Next, subtract:

The answer is 67.

### Examples

#### Example 1

Earlier, you were given a problem about Casey and his confusing homework.

Casey needs to expand the powers so he can see what operations he has to do to evaluate the expression.

First, evaluate the two powers given in the problem.

Next, replace the powers in the expression with their values.

The answer is 653.

#### Example 2

Evaluate the following variable expression when

First, substitute the value into the expression.

Next, expand the power.

Then, perform the operation in the parenthesis.

Multiply:

Next, multiply:

Then, subtract.

The answer is 116.

#### Example 3

Evaluate the following numerical expression.

First, expand the powers.

Next, write the expression in expanded form.

Then, in order from left to right multiply:

Next, multiply:

Then, in order from left to write, subtract:

\begin{align*}4+1+27\end{align*}

Next, add: \begin{align*}4+1=5\end{align*}and write the new expression.

\begin{align*}5+27\end{align*}

Then add: \begin{align*}5+27=32\end{align*}

The answer is 32.

#### Example 4

Evaluate the following variable expression when \begin{align*}m=3\end{align*}and\begin{align*}n=2\end{align*}.

\begin{align*}n^5+3m^2-15\end{align*}

First, substitute \begin{align*}m=3\end{align*}and\begin{align*}n=2\end{align*}into the variable expression.

\begin{align*}2^5+3(3)^2-15\end{align*}Next, expand the powers:\begin{align*}2^5=( 2 \times 2 \times 2 \times 2 \times 2) \ \text{and} \ (3)^2 =(3 \times 3)\end{align*}

Then, write the expression in expanded form.

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +3 (3 \times 3)-15\end{align*}

Then, perform the operation in the parenthesis.

First, multiply: \begin{align*}(3 \times 3)=(9)\end{align*}and write the new expression.

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 + 3(9)-15\end{align*}

Next, multiply: \begin{align*}3(9)=27\end{align*}to clear the parenthesis. Write the new expression.

\begin{align*}2 \times 2 \times 2 \times 2 \times 2 +27-15\end{align*}

Next, multiply: \begin{align*}2 \times 2 = 4 \times 2 = 8 \times 2 =16 \times 2 =32 \end{align*}Write the new expression.

\begin{align*}32+27-15\end{align*}

Next, add: \begin{align*}32+27=59\end{align*}and write the new expression.

\begin{align*}59-15\end{align*}Then, subtract: \begin{align*}59-15=44\end{align*}

The answer is 44.

### Review

Expand and evaluate each power.

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

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

3. \begin{align*}(-2)^4\end{align*}

4. \begin{align*}(-8)^2\end{align*}

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

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

7. \begin{align*}(-9)^2\end{align*}

8. \begin{align*}(-2)^6\end{align*}

Evaluate each numerical expression. Remember to apply PEMDAS to evaluate the expression accurately.

9. \begin{align*}6^2+22\end{align*}

10. \begin{align*}(-3)^3+18\end{align*}

11. \begin{align*}2^3+16-4\end{align*}

12. \begin{align*}(-5)^2-19\end{align*}

13. \begin{align*}(-7)^2+52-2\end{align*}

14. \begin{align*}18+9^2-3\end{align*}

15. \begin{align*}22-3^3+7\end{align*}

Evaluate each variable expression using the given values.

16. \begin{align*}6a+4^2-2\end{align*}, when \begin{align*}a=3\end{align*}

17. \begin{align*}a^3+14, \text{ when }a=6.\end{align*}

18. \begin{align*}2a^2-16, \text{ when }a=4\end{align*}

19. \begin{align*}5b^3+12, \text{ when }b= \text{-}2\end{align*}

20. \begin{align*}2x^2+52, \text{ when }x=4\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.6.

### Resources