For her best friend's birthday, Jean bought a huge birthday hat. It's too big to actually wear, so her plan is to add a base to it, buy fabric, and cover it with fabric. Jean needs to know how much fabric to buy. Jean's older brother told her that to find the approximate area of the surface of the birthday hat she can use the following formula:

where

is the diameter of the base of the hat and is the slant height of the hat.Jean measured her birthday hat and found that the diameter is 15 inches and the slant height is 18 inches. How can Jean use the formula that her brother gave her to find the surface area of the birthday hat so that she can buy the right amount of fabric?

In this concept, you will learn how to evaluate variable expressions involving exponents and parentheses.

### Evaluating Expressions with Powers

An **expression** is a mathematical phrase that contains numbers and operations. A **variable expression** is an expression that contains variables.

To **evaluate** a variable expression means to find the value of the expression for given values of the variables. To evaluate, substitute the given values for the variables in the expression and simplify using the order of operations (PEMDAS).

Remember that the order of operations are:

**P**arentheses: Start by simplifying any part of the expression in parentheses using the order of operations.**E**xponents: Rewrite any terms that contain exponents without exponents.**M**ultiplication/**D**ivision: Do any multiplication and/or division in order from left to right.**A**ddition/**S**ubtraction: Do any addition and/or subtraction in order from left to right.

Here is an example.

Evaluate the expression

if .First, substitute 4 in for

in the expression.

Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses. (4) in the first part of the expression is already simplified.

in the second part of the expression still needs to be simplified.

Next, continue to follow the order of operations and evaluate the exponent. Note that only the 4 is squared. Remember that means which is 16.

\begin{align*}2(4)^2-11=2(16)-11\end{align*}

Now, multiply.

Finally, subtract.

The answer is 21.

### Examples

#### Example 1

Earlier, you were given a problem about Jean and the birthday hat she bought for her friend's birthday. She wants to find the surface area of the hat so she can buy the correct amount of fabric to cover it. She got the following formula from her brother:

where

is the diameter of the base of the hat and is the slant height of the hat. Jean measured the birthday hat and found that the diameter is 15 inches and the slant height is 18 inches.To find the surface area, she should first substitute 15 in for

and 18 in for in the formula.\begin{align*}3.14\left ( \frac{15}{2} \right )^2+1.57(15)(18)\end{align*}

Next, notice that this expression has multiplication, division, exponents, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the \begin{align*}\left ( \frac{15}{2} \right )\end{align*}

Next, continue to follow the order of operations and evaluate the exponent. Use your calculator to help you calculate \begin{align*}7.5^2=7.5 \cdot 7.5\end{align*}

Now, multiply from left to right. You can use your calculator to help.

\begin{align*}\begin{array}{rcl}
3.14(56.25)+1.57(15)(18) & = & 176.625+1.57(15)(18)\\
& = & 176.625+23.55(18)\\
& = & 176.625+423.9
\end{array}\end{align*}

Finally, add.

\begin{align*}176.625+423.9=600.525\end{align*}

The answer is that the surface area of the hat (including the base that she added on) is about 600.525 square inches.

With that knowledge, Jean knows exactly how much fabric she needs to buy to help decorate the birthday hat!

#### Example 2

Evaluate

if is 4.First, substitute 4 in for

in the expression.

Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the \begin{align*}(4+3)\end{align*} at the end of the expression.

\begin{align*}6(4)^2-2(4)+(4+3)=6(4)^2-2(4)+7\end{align*}

Next, continue to follow the order of operations and evaluate the exponent.

Now, multiply from left to right.

Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.

The answer is 95.

#### Example 3

Evaluate

for .First, substitute 2 in for \begin{align*}x\end{align*} in the expression.

Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the

at the end of the expression.

Next, continue to follow the order of operations and evaluate the exponent.

Now, multiply.

Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.

The answer is 15.

#### Example 4

Evaluate

for and .First, substitute 3 in for

and 4 in for in the expression.

Next, notice that this expression has division, addition, parentheses, subtraction, and exponents. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the

in the middle of the expression.

Next, continue to follow the order of operations and evaluate the exponent.

Now, divide. Remember that a fraction bar means division so

is the same as .

Finally, add from left to right.

The answer is 30.

#### Example 5

Evaluate \begin{align*}5x^2-2+(3+3)\end{align*} if is 5.

First, substitute 5 in for the

in the expression.

Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the

at the end of the expression.

Next, continue to follow the order of operations and evaluate the exponent.

Now, multiply.

Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.

\begin{align*}\begin{array}{rcl} 125-2+6 & = & 123+6\\ & = & 129 \end{array}\end{align*}

The answer is 129.

### Review

Evaluate the following variable expressions for .

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Decide whether each of the following statements are true or false.

13. Parentheses are a grouping symbol.

14. Exponents can't be evaluated unless the exponent is equal to 3.

15. If there is multiplication and division in a problem you always do the multiplication first.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.10.