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1.10: Expression Evaluation with Powers and Grouping Symbols

Difficulty Level: At Grade Created by: CK-12
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Have you ever had a part time job?

Lydia and Bart both work at a bookstore. Lydia makes \begin{align*}x\end{align*} amount each hour; Bart has more experience, so he makes 1.5 times more than Lydia makes each hour. If Lydia and Bart both work 4 hours a day, how much will they make together in seven days if Lydia makes $8 an hour.

Write an expression; then solve it. Do you know how to do this?

This Concept will show you exactly how to do this.

Guidance

Now that we know the basic order of operations, we can expand our rules to include evaluation of more complicated expressions. In intricate expressions, parentheses are used as grouping symbols. Parentheses indicate which operations should be done first. In the order of operations, operations within the parentheses are always first.

Remember how to simplify and evaluate exponents?

\begin{align*}x^3 = xxx\end{align*}.

Exponential notation is another factor we must account for in the order of operations. After completing the operations in parentheses, we then evaluate the exponents. Then we complete the multiplication and division from left to right; finally, we complete the addition and subtraction from left to right. The chart below shows the complete order of operations. If you keep the order of operations in the forefront of your mind and are careful to take each step one at a time and show your work, evaluating complex expressions using the order of operations will become second nature.

Let’s look at the order of operation once again.

Evaluate the expression: \begin{align*}2x^2-(x+7)\end{align*} if \begin{align*}x=4\end{align*}

First, we substitute 4 in for \begin{align*}x\end{align*}.

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

Next, we add the terms in parentheses. Since parentheses is the first order of operations.

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

Now we can evaluate the exponent.

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

Multiplication is next and there is no division.

\begin{align*}32 - 11\end{align*}

Finally, we complete with subtraction since there is no addition.

21

Our answer is 21.

Sometimes you will also evaluate expressions with more than one variable. Just keep track and follow the order of operations and you will be all set.

Now it's time for you to practice.

Example A

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

Solution: 15

Example B

Evaluate \begin{align*}\frac{24}{x}+ (9-x)+y^2\end{align*} if \begin{align*}x\end{align*} is 3 and \begin{align*}y\end{align*} is 4

Solution: 30

Example C

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

Solution: 129

Now back to the dilemma at the bookstore. Here is the original problem once again.

Lydia and Bart both work at a bookstore. Lydia makes \begin{align*}x\end{align*} amount each hour; Bart has more experience, so he makes 1.5 times more than Lydia makes each hour. If Lydia and Bart both work 4 hours a day, how much will they make together in seven days if Lydia makes $8 an hour.

Write an expression; then solve it.

In this problem, \begin{align*}x\end{align*} stands for the amount Lydia makes per hour. Because Bart makes 1.5 times the amount that Lydia makes per hour, then \begin{align*}1.5x\end{align*} describes how much Bart makes per hour. The total Lydia and Bart make in 1 hour is therefore \begin{align*}(x + 1.5x)\end{align*}. Now the problem tells us that Lydia and Bart each work 4 hours per day. So, the amount they both make in 1 day is \begin{align*}4(x + 1.5x)\end{align*}. We want to find out how much they make in 7 days, so our expression is \begin{align*}7 \times 4(x + 1.5x)\end{align*}. The problem gives us the value of \begin{align*}x\end{align*}. Lydia makes $8 an hour. We substitute 8 for the variable in our expression and solve using the order of operations. Remember: parenthesis, exponents, multiplication, division, addition, subtraction.

\begin{align*}7 \times 4(x + 1.5x) &= 7 \times 4(8 + 1.5 \times 8)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(8 + 12)\\ 7 \times 4(x + 1.5x) &= 7 \times 4(20)\\ 7 \times 4(x + 1.5x) &= 560\end{align*}

The answer is $560.00.

Vocabulary

Here are the vocabulary words in this Concept.

Numerical Expression
an expression that uses numbers and operations.
Variable Expression
an expression that uses numbers, variables and operations.
Parentheses
grouping symbols, the first step of the order of operations.
Exponent
the little number that tells how many times to multiply the base times itself.
Order of Operations
the order that you perform each operation when evaluating an expression.

Guided Practice

Here is one for you to try on your own.

Evaluate \begin{align*}6x^2-2x+(x+3)\end{align*} if \begin{align*}x\end{align*} is 4

Answer

First, substitute 4 into every place that an \begin{align*}x\end{align*} appears.

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

Now evaluate according to the order of operations.

The answer is 95.

Video Review

Here is a video for review.

- This is a James Sousa video on evaluating an expressions using the order of operations.

Practice

Directions: Evaluate the following variable expressions is \begin{align*}x=4, y=2, z=3\end{align*}

1. \begin{align*}x^2+ y\end{align*}

2. \begin{align*}2y^2+ 5-2\end{align*}

3. \begin{align*}x^2- y^2+ z\end{align*}

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

5. \begin{align*}8+ x^2- 4y\end{align*}

6. \begin{align*}14 \div 2+ z^2- y\end{align*}

7. \begin{align*}20 + z^2-y\end{align*}

8. \begin{align*}5x-2y+3z\end{align*}

9. \begin{align*}5+(x-z)+ 5(6)\end{align*}

10. \begin{align*}8 + x-y^2+z\end{align*}

11. \begin{align*}(x+y)+ 5 \cdot 2 - 3\end{align*}

12. \begin{align*}4x^2+3z^3+ 2\end{align*}

Directions: Use what you have learned to answer the following questions 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 division first.

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Vocabulary

Evaluate

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

Exponent

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

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.

Parentheses

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

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

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