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# 1.2: Order of Operations

Difficulty Level: At Grade Created by: CK-12

## The Mystery of Math Verbs

Some math verbs are “stronger” than others and must be done first. This method is known as the Order of Operations.

A mnemonic (a saying that helps you remember something difficult) for the Order of Operations is PEMDAS - Please Excuse My Daring Aunt Sophie.

The Order of Operations:

Whatever is found inside PARENTHESES must be done first. EXPONENTS are to be simplified next. MULTIPLICATION and DIVISION are equally important and must be performed moving left to right. ADDITION and SUBTRACTION are also equally important and must be performed moving left to right.

Example 1: Use the Order of Operations to simplify \begin{align*}(7-2) \times 4 \div 2-3\end{align*}

Solution: First, we check for parentheses. Yes, there they are and must be done first.

\begin{align*}(7 - 2) \times 4 \div 2 - 3 = (5) \times 4 \div 2 - 3\end{align*}

Next we look for exponents (little numbers written a little above the others). No, there are no exponents so we skip to the next math verb.

Multiplication and division are equally important and must be done from left to right.

\begin{align*}5 \times 4 \div 2 - 3 & = 20 \div 2 - 3\\ 20 \div 2 - 3 & = 10 - 3\end{align*}

Finally, addition and subtraction are equally important and must be done from left to right.

\begin{align*}10-3 = 7\end{align*} This is our answer.

Example 2: Use the Order of Operations to simplify the following expressions.

a) \begin{align*}3 \times 5-7 \div 2\end{align*}

b) \begin{align*}3 \times (5 - 7) \div 2\end{align*}

c) \begin{align*}(3 \times 5) - (7 \div 2)\end{align*}

Solutions:

a) There are no parentheses and no exponents. Go directly to multiplication and division from left to right: \begin{align*} 3 \times 5 - 7 \div 2 = 15 - 7 \div 2 = 15 - 3.5\end{align*}

Now subtract: \begin{align*}15 - 3.5 = 11.5\end{align*}

b) Parentheses must be done first: \begin{align*}3 \times (-2) \div 2\end{align*}

There are no exponents, so multiplication and division come next and are done left to right: \begin{align*}3 \times (-2) \div 2 = -6 \div 2 = -3\end{align*}

c) Parentheses must be done first: \begin{align*}(3 \times 5) - (7 \div 2) = 15 - 3.5\end{align*}

There are no exponents, multiplication, division, or addition, so simplify:

\begin{align*}15 - 3.5 = 11.5\end{align*}

Parentheses are used two ways. The first is to alter the Order of Operations in a given expression, such as example (b). The second way is to clarify an expression, making it easier to understand.

Some expressions contain no parentheses while others contain several sets of parentheses. Some expressions even have parentheses inside parentheses! When faced with nested parentheses, start at the innermost parentheses and work outward.

Example 3: Use the Order of Operations to simplify \begin{align*}8-[19-(2+5)-7]\end{align*}

Solution: Begin with the innermost parentheses:

\begin{align*}8-[19-(2+5)-7]=8-[19-7-7]\end{align*}

Simplify according to the Order of Operations:

\begin{align*}8-[19-7-7]=8-[5]=3\end{align*}

## Evaluating Algebraic Expressions with Fraction Bars

Fraction bars count as grouping symbols for PEMDAS, and should be treated as a set of parentheses. All numerators and all denominators can be treated as if they have invisible parentheses. When real parentheses are also present, remember that the innermost grouping symbols should be evaluated first. If, for example, parentheses appear on a numerator, they would take precedence over the fraction bar. If the parentheses appear outside of the fraction, then the fraction bar takes precedence.

Example 4: Use the Order of Operations to simplify the following expressions.

a) \begin{align*}\frac{z + 3}{4} - 1\end{align*} when \begin{align*}z = 2\end{align*}

b) \begin{align*}\left (\frac{a+2}{b+4} - 1 \right ) + b\end{align*} when \begin{align*}a = 3\end{align*} and \begin{align*}b = 1\end{align*}

c) \begin{align*}2 \times \left ( \frac{w + (x - 2z)}{(y + 2)^2} - 1 \right )\end{align*} when \begin{align*}w = 11, \ x = 3, \ y = 1\end{align*} and \begin{align*}z = -2\end{align*}

Solutions: Begin each expression by substituting the appropriate value for the variable:

a) \begin{align*}\frac{(2+3)}{4} -1 = \frac{5}{4} -1\end{align*}. Rewriting 1 as a fraction, the expression becomes:

\begin{align*}\frac{5}{4} - \frac{4}{4} = \frac{1}{4}\end{align*}

b) \begin{align*}\frac{(3+2)}{(1+4)} = \frac{5}{5} = 1\end{align*}

\begin{align*}(1 - 1) + b\end{align*} Substituting 1 for b, the expression becomes \begin{align*} 0 + 1 = 1\end{align*}

c) \begin{align*}2 \left ( \frac{[11+(3-2(-2))]}{[(1+2)^2)]} - 1 \right ) = 2 \left ( \frac{(11+7)}{3^2} -1 \right ) = 2 \left (\frac{18}{9} - 1 \right )\end{align*}

Continue simplifying: \begin{align*}2\left ( \frac{18}{9} - \frac{9}{9} \right ) = 2 \left ( \frac{9}{9} \right ) = 2(1)= 2\end{align*}

## Using a Calculator to Evaluate Algebraic Expressions

A calculator, especially a graphing calculator, is a very useful tool in evaluating algebraic expressions. The graphing calculator follows the Order of Operations, PEMDAS. In this section, we will explain two ways of evaluating expressions with the graphing calculator.

Method #1: This method is the direct input method. After substituting all values for the variables, you type in the expression, symbol for symbol, into your calculator.

Evaluate \begin{align*}[3(x^2 - 1)^2 - x^4 + 12] + 5x^3 - 1\end{align*} when \begin{align*}x = -3\end{align*}.

Substitute the value \begin{align*}x = -3\end{align*} into the expression.

\begin{align*}[3((-3)^2 -1)^2 - (-3)^4 + 12] + 5(-3)^3 - 1\end{align*}

The potential error here is that you may forget a sign or a set of parentheses, especially if the expression is long or complicated. Make sure you check your input before writing your answer. An alternative is to type the expression in by appropriate chunks – do one set of parentheses, then another, and so on.

Method #2: This method uses the STORE function of the Texas Instrument graphing calculators, such as the TI-83, TI-84, or TI-84 Plus.

First, store the value \begin{align*}x = -3\end{align*} in the calculator. Type -3 [STO] \begin{align*}x\end{align*}. (The letter \begin{align*}x\end{align*} can be entered using the \begin{align*}x\end{align*}-[VAR] button or [ALPHA] + [STO]). Then type in the expression in the calculator and press [ENTER].

The answer is \begin{align*}-13.\end{align*}

Note: On graphing calculators there is a difference between the minus sign and the negative sign. When we stored the value negative three, we needed to use the negative sign, which is to the left of the [ENTER] button on the calculator. On the other hand, to perform the subtraction operation in the expression we used the minus sign. The minus sign is right above the plus sign on the right.

You can also use a graphing calculator to evaluate expressions with more than one variable.

Evaluate the expression: \begin{align*}\frac{3x^2 - 4y^2 + x^4}{(x + y)^{\frac{1}{2}}}\end{align*} for \begin{align*}x = -2, y = 1\end{align*}.

Store the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. \begin{align*}-2\end{align*} [STO] \begin{align*}x\end{align*}, 1 [STO] \begin{align*}y\end{align*}. The letters \begin{align*}x\end{align*} and \begin{align*}y\end{align*} can be entered using [ALPHA] + [KEY]. Input the expression in the calculator. When an expression shows the division of two expressions be sure to use parentheses: (numerator) \begin{align*}\div\end{align*} (denominator). Press [ENTER] to obtain the answer \begin{align*}-.8\bar{8}\end{align*} or \begin{align*}-\frac{8}{9}\end{align*}.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both. CK-12 Basic Algebra: Order of Operations (14:23)

Use the Order of Operations to simplify the following expressions.

1. \begin{align*}8 - (19 - (2 + 5) - 7)\end{align*}
2. \begin{align*}2 + 7 \times 11 - 12 \div 3\end{align*}
3. \begin{align*}(3 + 7) \div (7 - 12)\end{align*}
4. \begin{align*}\frac{2 \cdot (3 + (2 - 1))}{4 - (6 + 2)} - (3 - 5)\end{align*}
5. \begin{align*}8 \cdot 5 + 6^2\end{align*}
6. \begin{align*}9 \div 3 \times 7 - 2^3 + 7\end{align*}
7. \begin{align*}8 + 12 \div 6 + 6\end{align*}
8. \begin{align*}(7^2-3^2) \div 8\end{align*}

Evaluate the following expressions involving variables.

1. \begin{align*}\frac{jk}{j + k}\end{align*} when \begin{align*}j = 6\end{align*} and \begin{align*}k = 12\end{align*}.
2. \begin{align*}2y^2\end{align*} when \begin{align*}x = 1\end{align*} and \begin{align*}y = 5\end{align*}
3. \begin{align*}3x^2 + 2x + 1\end{align*} when \begin{align*}x = 5\end{align*}
4. \begin{align*}(y^2 - x)^2\end{align*} when \begin{align*}x = 2\end{align*} and \begin{align*}y = 1\end{align*}

Evaluate the following expressions involving variables.

1. \begin{align*}\frac{4x}{9x^2 - 3x + 1}\end{align*} when \begin{align*}x = 2\end{align*}
2. \begin{align*}\frac{z^2}{x + y} + \frac{x^2}{x - y}\end{align*} when \begin{align*}x = 1, \ y = -2\end{align*}, and \begin{align*}z = 4\end{align*}.
3. \begin{align*}\frac{4xyz}{y^2 - x^2}\end{align*} when \begin{align*}x = 3, \ y = 2\end{align*}, and \begin{align*}z = 5\end{align*}
4. \begin{align*}\frac{x^2 - z^2}{xz - 2x(z - x)}\end{align*} when \begin{align*}x = -1\end{align*} and \begin{align*}z = 3\end{align*}

The formula to find the volume of a square pyramid is \begin{align*}V=\frac{s^2 (h)}{3}\end{align*}. Evaluate the volume for the given values.

1. \begin{align*}s=4\ inches,h=18\ inches\end{align*}
2. \begin{align*}s=10\ feet,h=50\ feet\end{align*}
3. \begin{align*}h=7\ meters,s=12\ meters\end{align*}
4. \begin{align*}h=27\ feet,s=13\ feet\end{align*}
5. \begin{align*}s=16\ cm,h=90\ cm\end{align*}

In 22 – 25, insert parentheses in each expression to make a true equation.

1. \begin{align*}5 - 2 \cdot 6 - 4 + 2 = 5\end{align*}
2. \begin{align*}12 \div 4 + 10 - 3 \cdot 3 + 7 = 11\end{align*}
3. \begin{align*}22 - 32 - 5 \cdot 3 - 6 = 30\end{align*}
4. \begin{align*}12 - 8 - 4 \cdot 5 = -8\end{align*}

In 26 – 29, evaluate each expression using a graphing calculator.

1. \begin{align*}x^2 + 2x - xy\end{align*} when \begin{align*}x = 250\end{align*} and \begin{align*}y = -120\end{align*}
2. \begin{align*}(xy - y^4)^2\end{align*} when \begin{align*}x = 0.02\end{align*} and \begin{align*}y = -0.025\end{align*}
3. \begin{align*}\frac{x + y - z}{xy + yz + xz}\end{align*} when \begin{align*}x = \frac{1}{2}, \ y = \frac{3}{2}\end{align*}, and \begin{align*}z = -1\end{align*}
4. \begin{align*}\frac{(x + y)^2}{4x^2 - y^2}\end{align*} when \begin{align*}x = 3\end{align*} and \begin{align*}y = -5d\end{align*}
5. The formula to find the volume of a spherical object (like a ball) is \begin{align*}V = \frac{4}{3}(\pi)r^3\end{align*}, where \begin{align*}r =\end{align*} the radius of the sphere. Determine the volume for a grapefruit with a radius of 9 cm.

Mixed Review

1. Let \begin{align*}x = -1\end{align*}. Find the value of \begin{align*}-9x + 2\end{align*}.
2. The area of a trapezoid is given by the equation \begin{align*}A = \frac{h}{2}(a + b)\end{align*}. Find the area of a trapezoid with bases \begin{align*}a = 10 \ cm, b = 15 \ cm\end{align*}, and height \begin{align*}h = 8 \ cm\end{align*}.
3. The area of a circle is given by the formula \begin{align*}A = \pi r^2\end{align*}. Find the area of a circle with radius \begin{align*}r = 17\end{align*} inches.

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