# Order of Operations

## The order of operations outlines the order by which certain operatives should take place in a mathematical expression for it to remain true

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Order of Operations with Negative Real Numbers

Ginny walked into Math class and saw the following question on the board:

Find the value of the following expression when \begin{align*}a = -2, b = 3, c = -1, d = 1\end{align*}:
\begin{align*}\boxed{(4a+c)\div b-(bd)\div (ac)}\end{align*}

### Order of Operations with Negative Numbers

The standard order of operations involves specific steps for performing the mathematical calculations presented in a mathematical statement. These steps are represented by the letters PEMDAS.

P – Parentheses – Do all the calculations within parentheses.

E – Exponents – Do all calculations that involve exponents.

M/D – Multiplication/Division – Do all multiplication and division, in the order it occurs, working from left to right.

A/S – Addition/Subtraction – Do all addition and subtraction, in the order it occurs, working from left to right.

These steps do not change whether they are being applied to positive real numbers or to negative real numbers. The rules for adding, subtracting, multiplying and dividing real negative numbers must be applied when evaluating expressions that require PEMDAS to be used. A good way to learn the order of operations is to observe examples where it is correctly applied. The majority of this lesson is a number of problems that you should follow step-by-step in order to help cement the order in your memory.

Let's complete the following problems using PEMDAS:

1. Simplify the expression: \begin{align*}32 \div 4^2 \times 2-21\end{align*}

There are no calculations inside parentheses. The first step is to evaluate the number with the exponent.

\begin{align*}=32\div {\color{blue}16}\times 2-21\end{align*}

The next step is to perform any division or multiplication, working from left to right.

\begin{align*}&={\color{blue}2} \times 2-21 \\ &={\color{blue}4}-21\end{align*}

The final step is to rewrite the expression as an addition problem and to change the sign of the original number being subtracted.

\begin{align*}=4{\color{blue}+}(\text{-}21 )\end{align*}

When adding two numbers with unlike signs, subtract the smaller number from the larger, then the sign of the number with the greater magnitude will be the sign of the answer.

\begin{align*}= {\color{blue}\text{-}17}\end{align*}

1. Find the value of the following expression when \begin{align*}a = \text{-}2, b = 3, c = \text{-}1, \text{ and }d = 1.\end{align*}

\begin{align*}(4a^2c^2)-(3ac^3)\end{align*}

Begin by substituting the variables with the given values. Use brackets to group the operations with parentheses.

\begin{align*}=[4(\text{-}2)^2(\text{-}1)^2] - [3(\text{-}2)(\text{-}1)^3]\end{align*}

In the first set of brackets, do the calculations with exponents.

\begin{align*}=[4({\color{blue}4})({\color{blue}1})] - [3(\text{-}2)(\text{-}1)^3] \end{align*}

In the second set of brackets, do the calculations with exponents.

\begin{align*}=[4(4)(1)]-[3(\text{-}2)({\color{blue}\text{-}1})] \end{align*}

In the first set of brackets, do the multiplication. The brackets can now be removed.

\begin{align*}={\color{blue}16}-[3(\text{-}2)({\color{blue}\text{-}1})] \end{align*}

In the second set of brackets, do the multiplication. The brackets can now be removed.

\begin{align*}=16-{\color{blue}6} \end{align*}

Subtract the numbers.

\begin{align*}={\color{blue}10}\end{align*}

1. What is the value of \begin{align*}3-2 \left[\frac{8(\text{-}1)-7}{\text{-}3(2)-4}\right]\end{align*}?

Simplify within the brackets first:
\begin{align*}&=3-2\left[\frac{{\color{blue}\text{-}8}-7}{\text{-}3(2)-4}\right] \\ &=3-2\left[\frac{\text{-}8-7}{{\color{blue}\text{-}6}-4}\right]\end{align*}

Rewrite subtraction as addition of the opposite:

\begin{align*}&=3-2 \left[\frac{\text{-}8{\color{blue}+}(\text{-}7)}{\text{-}6{\color{blue}+}(\text{-}4)}\right]\\ &=3-2\left[\frac{\text{-}8-7}{{\color{blue}\text{-}6}-4}\right]\\ &=3-2\left[{\color{blue}\frac{\text{-}15}{\text{-}10}}\right] \\ &=3- {\color{blue}\frac{\text{-}30}{\text{-}10}}\\ &=3-{\color{blue}3}\\ &={\color{blue}0}\end{align*}

### Examples

#### Example 1

Earlier, you read about the math question that Ginny saw written on the board in her math class:

Find the value of the following expression when \begin{align*}a = -2, b = 3, c = -1, d = 1\end{align*}:

\begin{align*}\boxed{(4a+c)\div b-(bd)\div (ac)}\end{align*}

Ginny felt good about answering the problem because she remembered the steps involved in the standard order of operations. When she noticed that the values for two of the variables were negative numbers, she realized that she would have to be careful doing the calculations because she would also have to apply the rules that she had learned for adding, subtracting, multiplying and dividing negative real numbers.

\begin{align*}(4a+c)\div b-(bd) \div(ac)\end{align*}

Ginny began by substituting the variables with the given values.

\begin{align*}(4(-2)+(-1))\div 3-((3)(1))\div((-2)(-1))\end{align*}

To reduce errors in her calculations, Ginny wrote all of the values in parentheses. The statement now has parentheses within parentheses. This may seem confusing and the order in which to perform the operations may become skewed. Ginny asked her teacher about writing the expression another way. Her teacher advised her to replace the outer parentheses with brackets [ ]. Brackets are another type of grouping symbol. When evaluating an expression that has grouping symbols (parentheses) within grouping symbols (brackets), perform the operations within the innermost set of symbols first.

Ginny rewrote the expression using both brackets and parentheses.

\begin{align*}[4(-2)+(-1)]\div 3-[(3)(1)] \div [(-2)(-1)]\end{align*}

Then it was easier to keep track of the steps and the proper order to simplify the expression:

\begin{align*}&=[{\color{blue}-8}+(-1)]\div 3 - [(3)(1)] \div [(-2)(-1)]\end{align*}

\begin{align*}&=[\text{-}8+(\text{-}1)] \div 3-[3]\div[2]\\ &=[{\color{blue}\text{-}9}] \div 3-[3] \div [2]\\ &=\text{-}9\div 3-3\div 2\\ &={\color{blue}\text{-}3}-3 \div 2\\ &=\text{-}3- \color{blue}1.5 \color{black}\rightarrow \text{-}3+ \text{-}1.5\\ &={\color{blue}\text{-}4.5}\end{align*}

#### Example 2

Perform the following operations: \begin{align*}8 \times -9 +19 \div (-30+11)-14 \times (-1)^2\end{align*}

\begin{align*} & 8 \times -9+19 \div (-30+11)-14 \times (-1)^2\\ & =8 \times 9 +19 \div {\color{blue}-19}-14 \times (-1)^2 \\ & =8 \times9+19 \div {\color{blue}-19}-14 \times (-1)^2 \\ & ={\color{blue}72}+19 \div -19-14\times 1\\ & =72+{\color{blue}-1}-14 \times 1 \\ & =72+-1-{\color{blue}14} \\ & ={\color{blue}71}-14 \\ & ={\color{blue}57}\end{align*}

#### Example 3

Simplify: \begin{align*}\left(\frac{-12-6}{6+3}\right)+\left(\frac{-36}{-4}\right)+(-8 \times 2)\end{align*}

\begin{align*}& \left(\frac{-12-6}{6+3}\right)+\left(\frac{-36}{-4}\right)+(-8 \times 2)\\ & =\left({\color{blue}\frac{-18}{9}}\right)+\left(\frac{-36}{-4}\right)+(-8 \times 2)\\ & ={\color{blue}-2}+\left(\frac{-36}{-4}\right)+(-8 \times 2)\\ & =-2+{\color{blue}9}+(-8 \times 2)\\ & =-2+9+{\color{blue}-16}\\ & ={\color{blue}-9}\end{align*}

#### Example 4

A formula from geometry is \begin{align*}V=\frac{h}{6}(B+4M+b)\end{align*}. Find \begin{align*}V\end{align*} when \begin{align*}h = -15, B = 12,M = 8, b = 4\end{align*}.

\begin{align*}& V=\frac{h}{6}(B+4M+b).\\ & V=\frac{-15}{6}(12+4(8)+4)\\ & V=\frac{-15}{6}(12+{\color{blue}32}+4)\\ & V=\frac{-15}{6}(\color{blue}48)\\ & V={\color{blue}-2.5}(48)\\ & V={\color{blue}-120}\end{align*}

### Review

1. If \begin{align*}a = -3, b = 1\end{align*} and \begin{align*}c = 2\end{align*}, what is the value of \begin{align*}2a^3-3b+2c^2\end{align*}? _______
1. 221
2. 59
3. –49
4. 43
2. Evaluate \begin{align*}a-bc\end{align*} when \begin{align*}a=\frac{1}{2}, b=\frac{1}{3}\end{align*} and \begin{align*}c=\frac{5}{4}\end{align*}:  _______
1. 1
2. \begin{align*}\frac{1}{12}\end{align*}
3. \begin{align*}\frac{2}{5}\end{align*}
4. \begin{align*}\frac{5}{2}\end{align*}
3. Evaluate: \begin{align*}a(-b^2-a^2)\end{align*} when \begin{align*}a=-3\end{align*} and \begin{align*}b=4\end{align*}:  _______
1. 75
2. 2
3. 3
4. 4
4. Simplify the following: \begin{align*}[3-[5-(6-8)]+4]-2\end{align*} = _______
1. –6
2. –2
3. 2
4. –19
5. Perform the following operations and evaluate: \begin{align*}(5\times 3-7)^2 \div 4+9\end{align*} = _______
1. 109
2. 5
3. 11
4. 25

Perform the indicated calculations.

6. \begin{align*}\frac{6-(24-14)}{-10-[2-(-4)^2]}\end{align*}
7. \begin{align*}\frac{[12-(-15+6)]\times 4 -16}{-4}\end{align*}
8. \begin{align*}3\left(-2 \times \frac{20}{-4 \times -1}-5-7\right)\end{align*}
9. \begin{align*}-3 \times 5 -[3(3+9)-9\times -3]\end{align*}
10. \begin{align*}4^2-(4+8-2-4-2\times 7)\end{align*}
11. \begin{align*}2\times -18+9\div -3-5\times -2\end{align*}
12. \begin{align*}-4\times (-12-8)\div -2\end{align*}
13. \begin{align*}-15-6\div 3\end{align*}
14. \begin{align*}\frac{-2-(15-5)}{-4\times -2-12+-6\div 3}\end{align*}
15. Which expression has the greatest value if \begin{align*}a=-2\end{align*} and \begin{align*}b=3\end{align*}?
1. \begin{align*}3[a^2+b^2-2ab-2(a^2-b^2)]\end{align*}
2. \begin{align*} 3(a-b)-2(b-a)+3(a-2b)\end{align*}
3. \begin{align*} b(a-b)-a(b-a)-ab\end{align*}

To see the Review answers, open this PDF file and look for section 1.12.

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### Vocabulary Language: English

TermDefinition
Brackets Brackets [ ], are symbols that are used to group numbers in mathematics. Brackets are the 'second level' of grouping symbols, used to enclose items already in parentheses.
Grouping Symbols Grouping symbols are parentheses or brackets used to group numbers and operations.
nested parentheses Nested parentheses describe groups of terms inside of other groups. By convention, nested parentheses may be identified with other grouping symbols, such as the braces "{}" and brackets "[]" in the expression $\{ 3 + [ 2 - ( 5 + 4 ) ] \}$. Always evaluate parentheses from the innermost set outward.
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
PEMDAS PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.