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Numerical Expression Evaluation with Grouping Symbols

Practice applying the order of operations

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Numerical Expression Evaluation with Grouping Symbols
License: CC BY-NC 3.0

Doug is collecting all the ribbon he can find in his house. He receives three 6 ft blue ribbon rolls from his mom, finds 4 ft of orange ribbon in a drawer, and grabs 7 ft of gold ribbon from his little sister, who then comes over with scissors and takes back 2 ft for herself. Doug sits at his desk and starts to measure all his collected ribbon with a measuring tape. Is there an easier way for Doug to figure out how many feet of ribbon he has?

In this concept, you will learn how to evaluate numerical expressions using powers and grouping symbols.

Order of Operations

Parentheses are symbols that group things together. This becomes very important in numerical expressions, because operations inside parentheses are always completed first when evaluating the expression. Let’s review the order of operations.

 Order of Operations

P - parentheses

E - exponents

MD - multiplication or division, in order from left to right

AS - addition or subtraction, in order from left to right

You can see that, according to the order of operations, parentheses come first. 

Let’s see how this works.

\begin{align*}2 + (3 - 1) \times 2\end{align*}

In this problem, there are four elements to consider. You have one set of parentheses, addition, subtraction and multiplication. You can evaluate this expression using the order of operations. Here is what the process looks like:.

\begin{align*}& 2 + (3 - 1) \times 2\\ & 2 + 2 \times 2\\ & 2 + 4\\ & = 6\end{align*}

The answer is 6.

Let's consider a different problem, and take it step by step:

\begin{align*}35 + 3^2 - (3 \times 2) \times 7\end{align*}

First, evaluate parentheses.

\begin{align*} 35 + 3^2 - 6 \times 7\end{align*}

Next, evaluate exponents.

\begin{align*}35 + 9 - 6 \times 7\end{align*}

Then, complete multiplication in order from left to right.

\begin{align*}35 + 9 - 42\end{align*}

Finally, complete addition and/or subtraction in order from left to right.

\begin{align*}44 - 42 = 2\end{align*}

The answer is 2.

Examples

Example 1

Earlier, you were given a problem about Doug and his ribbon pile.

Doug needs to figure out the total length of ribbon he has collected.

He receives 3 of the 6 ft blue ribbon rolls from his mom, finds 4 ft of orange ribbon in a drawer, and grabs 7 ft of gold ribbon from his little sister, who then comes over with scissors and takes back 2 ft for herself.

First, identify the important information.

receives 3 6 ft rolls

finds 4 ft

takes 7 ft

gives back 2 ft

Next, write this as an expression.

\begin{align*}3 \times 6 + 4 + 7 - 2\end{align*}

Then, use order of operations to evaluate the expression.

\begin{align*}3 \times 6 + 4 + 7 - 2 & \quad \text{Multiply }3 \times 6 = 18\\ 18+4+7-2 & \quad \text{Add }18+4=22\\ 22+7-2& \quad \text{Add }22+7=29\\ 29-2 & \quad \text{Subtract }29-2=27\\ 27\\\end{align*}

The answer is 27.

Doug has collected 27 feet of ribbon with which to make mischief.

Example 2

Evaluate the following expression.

\begin{align*}7^3 - 3^2 + 15 \times 2 + (2 + 3)\end{align*}

First, follow the order of operations and evaluate the parentheses and exponents.

\begin{align*}{7}^{3}=7\times 7\times 7= &343\\ {3}^{2}=3\times 3= &9\\ \left(2+3\right)= &5\end{align*} 

Next, substitute these values back into the original number sentence.

\begin{align*}343 - 9 + 15 \times 2 + 5\end{align*}

Then, complete the multiplication.

\begin{align*}15\times 2=30\end{align*}

Finally, complete the addition and subtraction in order from left to right.

\begin{align*}343 - 9 + 30 + 5\end{align*}

\begin{align*}369\end{align*}

The answer is 369.

Example 3

Evaluate the following expression.

\begin{align*}16 + 2^3 - 5 + (3 \times 4)\end{align*}

First, evaluate the operations inside of parenthesis. 

\begin{align*}16+2^3-5+(3 \times 4) & \quad \text{Evaluate }3 \times 4 = 12\\\end{align*}

Next, evaluate the exponents.

\begin{align*}16+2^3-5+12& \quad \text{Evaluate }2^3=8 \\ 16+8-5+12&\\ \end{align*}

Then, complete the addition and subtraction from left to right.

\begin{align*}16+8-5+12 & \quad \text{Add }16+8=24\\ 24-5+12 & \quad \text{Subtract }24-5=19\\ 19+12 & \quad \text{Add }19+12= 31\\ 31 &\\\end{align*}

The answer is 31.

Example 4

Evaluate the following expression.

\begin{align*}9^2 + 2^2 - 5 \times (2 + 3)\end{align*}First, evaluate the parenthesis.

 \begin{align*}9^2+2^2-5 \times (2+3) & \quad \text{Evaluate }2+3=5\\ 9^2+2^2-5 \times 5 & \\\end{align*}

Next, evaluate the exponents.

\begin{align*}9^2+2^2-5 \times 5 & \quad \text{Evaluate }9^2=81\\ 81+2^2-5 \times 5 & \quad \text{Evaluate }2^2 =4\\ 81+4-5 \times 5\\\end{align*}

Then, multiply.

\begin{align*}81+4-5 \times 5 & \quad \text{Multiply }5 \times 5=25\\ 81+4-25\\\end{align*}

Finally, complete the addition and subtraction operations from left to right.

\begin{align*}81+4-25 & \quad \text{Add }81+4=85\\ 85-25 & \quad \text{Subtract }85-25=60\\ 60 &\\\end{align*}

The solution is 60.

Example 5

Evaluate the following expression.

\begin{align*}8^2 \div 2 + 4 - 1 \times 6\end{align*}

First, evaluate the exponent.

\begin{align*}8^2 \div 2+4-1 \times 6 & \quad \text{Evaluate }8^2=64\\ 64 \div 2+4-1 \times 6 &\\\end{align*}

Then, divide and multiply from left to right.

\begin{align*}64 \div 2+4-1 \times 6 & \quad \text{Divide }64 \div 2 = 32\\ 32+4-1 \times 6 & \quad \text{Multiply }1 \times 6 = 6\\ 32+4-6 &\\\end{align*}Next, add and subtract from left to right.

 \begin{align*}32+4-6 & \quad \text{Add }32+4=36\\ 36-6 & \quad \text{Subtract }36-6=30\\ 30\\\end{align*}

The answer is 30.

Review

Evaluate each expression according to the order of operations.

  1. \begin{align*}3+(2+7)-3+5 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  2. \begin{align*}2+(5-3)+7^2-11 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  3. \begin{align*}4\times2+(6-4)-9+5 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  4. \begin{align*}8^2-4+(9-3)+12 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  5. \begin{align*}7^3-100+(3+4)-9 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  6. \begin{align*}7+(3^2+7)-11+5 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  7. \begin{align*}2^4+(8+7)+13-5 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  8. \begin{align*}3\times2+(2^2+7)-11+15 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  9. \begin{align*}8+(6+7)-2\times3 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  10. \begin{align*}22+(3^4+7)-73+15 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  11. \begin{align*}3^2+(4^2-7)-3+25 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  12. \begin{align*}6^3+(3^2+17)-73+4 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  13. \begin{align*}243-(5^3+27)-83+9 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  14. \begin{align*}72 + (11^2+117)-193+75 = \underline{\;\;\;\;\;\;\;}\end{align*} 
  15. \begin{align*}82 + (10^2+130)-303+115 = \underline{\;\;\;\;\;\;\;}\end{align*}  

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.11. 

Resources

 

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Vocabulary

Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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