3.2 Order of Operations

Narrated lesson: Order of Operations

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Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first.

You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.”

Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations, arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, $4 \times 2^{2} + 3 + 5^{2}$ can be calculated in many ways. You could get 5,184. Or, you could get 80. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find $4 \times 2^{2} + 3 + 5^{2} = 44$ .

Simplify Expressions Using Order of Operations

The order in which mathematical operations are performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS:

E	Exponents
M/D	Multiplication and division
A/S	Addition and Subtraction

So, what does EMDAS tell us to do? In an expression, first evaluate the separately grouped powers shown in this lesson. For chained powers, use parentheses to make their grouping explicit. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example.

Your Turn

Calculate $43+18\times15$ .

Show solution

$43 + 18 \times 15$

Use PEMDAS. The highest order operation in this expression is multiplication.

Multiply.	$43 + 270$
Add.	$313$

313

Your Turn

Calculate ${60^2}/50$ .

Show solution

${60^2}/50$

Use PEMDAS.

Calculate the exponent.	$3600/50$
Divide.	$72$

Your Turn

Calculate $3\times{6^3}-18$ .

Show solution

$3 \times {6^3}-18$

Use PEMDAS.

Calculate the exponent.	$3 \times 216-18$
Multiply.	$648-18$
Subtract.	$630$

630

Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS.

Example 4 Using Eight Order of Operations

Correctly apply the order of operations to compute the following:

$4 - 25 \times 6 / 10 \times 3^{2} + 7 \times 2^{3}$ .

Show solution

Step 1: To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the expression, 3 squared and 2 cubed.

$4 - 25 \times 6 / 10 \times 3^{2} + 7 \times 2^{3} = 4 - 25 \times 6 / 10 \times 9 + 7 \times 8$

Step 2: Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is the product of 25 and 6.

$\begin{array}{l} 4 - 25 \times 6 / 10 \times 9 + 7 \times 8 \\ = 4 - 150 / 10 \times 9 + 7 \times 8 \end{array}$

Step 3: Next is the 150 divided by 10.

$\begin{array}{l} 4 - 150 / 10 \times 9 + 7 \times 8 \\ = 4 - 15 \times 9 + 7 \times 8 \end{array}$

Step 4: Next is 15 multiplied by 9.

$\begin{array}{l} 4 - 15 \times 9 + 7 \times 8 \\ = 4 - 135 + 7 \times 8 \end{array}$

Step 5: Finally, multiply the 7 and 8.

$\begin{array}{l} 4 - 135 + 7 \times 8 \\ = 4 - 135 + 56 \end{array}$

As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left to right.

$\begin{array}{l} 4 - 135 + 56 \\ = - 131 + 56 \\ = - 131 + 56 \\ = - 75 \end{array}$

The computed value is −75.

Your Turn

Correctly apply the order of operations to compute the following:

$13+{3^4}\times8/6\times{5^2}-3\times4$ .

Show solution

$13 + {3^4} \times 8/6 \times {5^2}-3 \times 4$

Use PEMDAS.

Calculate the exponents.	$13 + 81 \times 8/6 \times 25-3 \times 4$
Multiply times 8.	$13 + 648/6 \times 25-3 \times 4$
Divide by 6.	$13 + 108 \times 25-3 \times 4$
Multiply by 25.	$13 + 2,700-3 \times 4$
Multiply by 4.	$13 + 2,700-12$
Add.	$2,713-12$
Subtract.	$2,701$

2,701

Your Turn

Correctly apply the rules for the order of operations to accurately compute the following:

$12/(-4)+8\times9/12\times{2^3}-24\times25/10$ .

Show solution

$12/\left({-4} \right) + 8 \times 9/12 \times {2^3}-24 \times 25/10$

Use PEMDAS.

Calculate the exponent.	$12/\left({-4} \right) + 8 \times 9/12 \times 8-24 \times 25/10$
Divide by –4.	$-3 + 8 \times 9/12 \times 8-24 \times 25/10$
Multiply by 9.	$-3 + 72/12 \times 8-24 \times 25/10$
Divide by 12.	$-3 + 6 \times 8-24 \times 25/10$
Multiply by 8.	$-3 + 48-24 \times 25/10$
Multiply by 25.	$-3 + 48-600/10$
Divide by 10.	$-3 + 48-60$
Add.	$45-60$
Subtract.	$-15$

Your Turn

Correctly apply the rules for the order of operations to accurately compute the following:

$3\times{4^3}\times7+24/6\times{7^2}-9/3\times8$ .

Show solution

$3 \times {4^3} \times 7 + 24/6 \times {7^2}-9/3 \times 8$

Use PEMDAS.

Calculate the exponents.	$3 \times 64 \times 7 + 24/6 \times 49-9/3 \times 8$
Multiply times 64.	$192 \times 7 + 24/6 \times 49-9/3 \times 8$
Multiply times 7.	$1,344 + 24/6 \times 49-9/3 \times 8$
Divide by 6.	$1,344 + 4 \times 49-9/3 \times 8$
Multiply by 49.	$1,344 + 196-9/3 \times 8$
Divide by 3.	$1,344 + 196-3 \times 8$
Multiply by 8.	$1,344 + 196-24$
Add.	$1,540-24$
Subtract.	$1,516$

1,516

Using the Order of Operations Involving Grouping Symbols

We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below.

P	Parentheses
E	Exponents
M/D	Multiplication and division	(division by a nonzero number is multiplication by its reciprocal)
A/S	Addition and subtraction	(subtraction is just the addition of the negative)

As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: $open parenthesis 4 times 7 close parenthesis squared$ .

This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS.

Your Turn

Correctly apply the rules for the order of operations to accurately compute the following:

$8-(25-{2^2})/7$ .

Show solution

$8-\left({25-{2^2}} \right)/7$

Use PEMDAS.

Evaluate the operations inside the parentheses first.

Calculate the exponent.	$8-\left({25-4} \right)/7$
Subtract inside the parentheses.	$8-\left({21} \right)/7$
Divide by 7.	$8-3$
Subtract.	$5$

Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward.

Example 8 Working Innermost Parentheses in the Order of Operations

Correctly apply the rules for order of operations to accurately compute the following:

$4 + 2 \times (3^{2} - {(2 + 5)}^{2} \times 4) / (3 + 8)$ .

Show solution

Step 1: Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5.

$\begin{array}{l} 4 + 2 \times (3^{2} - {(2 + 5)}^{2} \times 4) / (3 + 8) \\ = 4 + 2 \times (3^{2} - {(7)}^{2} \times 4) / (11) \end{array}$

Step 2: Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the subtraction.

$\begin{array}{l} 4 + 2 \times (3^{2} - 7^{2} \times 4) / 11 \\ = 4 + 2 \times (3^{2} - 49 \times 4) / 11 \\ = 4 + 2 \times (- 187) / 11 \end{array}$

Step 3: Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation.

$4 + 2 \times (- 187) / 11 = 4 - 374 / 11 = 4 - 34 = - 30$

Your Turn

Correctly apply the rules for the order of operations to accurately compute the following:

${(8-6)^2}\times100-({(48/6-3)^2}-4\times7)$ .

Show solution

${({8-6})^2} \times 100-({{{({48/6-3})}^2}-4 \times 7})$

Use PEMDAS.

Evaluate the operations inside the parentheses first.

Subtract inside the first parentheses.	${(2)^2} \times 100-({{{({48/6-3})}^2}-4 \times 7})$
Divide by 6 inside the inner parentheses.	${(2)^2} \times 100-({{{({8-3})}^2}-4 \times 7})$
Subtract inside the inner parentheses.	${(2)^2} \times 100-({{{(5)}^2}-4 \times 7})$
Evaluate the exponent inside the second parentheses.	${(2)^2} \times 100-({25-4 \times 7})$
Multiply by 7 inside the second parentheses.	${(2)^2} \times 100-({25-28})$
Subtract inside the second parentheses.	${(2)^2} \times 100-({-3})$
Evaluate the exponent.	$4 \times 100-({-3})$
Multiply by 100.	$400-({-3})$
Subtract.	$403$

403

Which operations have the lowest precedence in order of operations?

In an expression containing addition and subtraction at the same precedence, or multiplication and division at the same precedence, in what order should those operations be performed? Chained powers are a separate case: use parentheses to make their grouping explicit.

Which operations have the same precedence in order of operations?

After all operations in parentheses have been performed, which operations should be performed next?

For the following exercises, perform the indicated calculation.

$4-5\times6$

$34-10\times6$

$18+8\times13$

$40+12\times17$

$50\div2+8$

$72\div6+18$

$13+4\times{3^2}$

$45-6\times{7^3}$

${6^2}\times5-13\times{9^2}$

${14^2}\times8-5\times{2^4}$

$450\div{3^2}-56 \div {2^2}$

$\text{1,000} \div {5^3}-7 \times {8^4}$

$38 \times 6-4+5 \times 18 \div 10$

$15 \times 7+23-6 \times 40 \div 24$

$600 \div 12 \times 5+40-{6^2}$

$2 \times {12^3} \div 8 \times 3-{4^5}$

$10 \times {6^3} \div 2 \times 5+{3^2}-240 \div 8 \times 9$

$45 \div 15 \times 6+{7^2}-4 \times 15 \times 18 \div 12 \times 3$

$(4+3) \times 2$

$6 \times (12+8)$

$(14-25) \times 2$

$(86-61) \div 5$

${(3+2)^3}$

${(17 - 12)^4}$

${(45-60)^2}$

${(90-101)^3}$

${(3 \times 4)^3} \div 8+5$

${(6-9)^2} \times 5-4$

$4-(6+3) \times 5+11$

$12+(13-6) \times 8-130$

$15 \times {(4-1)^2}+5 \times (17+2 \times 3)+4$

$18 \times {(12-6)^4}-8 \times (30-14 \times 16)-90$

$4 \times (5+2 \times (6+8))+10 \times {3^2}$

$9 \times (13-12 \times (41-32))-8 \times 45$

$21-3 \times (5 \times (2+6)-3 \times (18-11)+25) \div 2$

$48-6 \times (10 \times (18+12)-25 \times (16-9)+19) \div 4$

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