Math 6 · 6.EE.A.1 · 6.EE.A.2C

Order of operations with powers

In 2 + 3 squared, is the 2 squared as well? When an expression mixes powers with other operations, which part is worked out first, and why that one?

You need

Grouping symbolsGrade 5 Base, exponent and the value of a powerMath 6

Unlocks

Order of operations with negative numbersMath 7

Where you are 5 in Order of operations

Number9

Arithmetic5

Order of operations5

Numerical expressions Grouping symbols

Order of operations with powers

Writing and reading numerical expressions from words Order of operations with negative numbers

When you reach for it

Reach for this whenever an expression holds a power together with other operations, such as 2 + 3^2, 4 \times 5^3 - 1 or a fraction with powers on its top and bottom.

Written down, 2 + 3^2 could be read two ways. Work the power first and it is 2 + 9 = 11. Add first and it is 5^2 = 25. A written expression is only useful if everyone who reads it gets the same number, so there has to be an agreed order. The agreed order is a convention, a choice people made, but it was not chosen at random: each step of it has a reason, and the reasons are what this lesson is about. What a power is and how to find its value is in Base, exponent and the value of a power.

The procedure

The order, from first to last:

grouping symbols: brackets, fraction bars, and anything raised as an exponent;
powers;
multiplying and dividing, as one level, from left to right;
adding and subtracting, as one level, from left to right.

Inside a grouping symbol the same order applies again. Some people remember the levels by a word built from their first letters; the reasons below are what make the order hold together.

Why powers come before multiplying

A power is short for a product. 2 \times 3^2 means 2 \times 3 \times 3, which is 18 whatever order its factors are multiplied in. Working the power first agrees: 2 \times 9 = 18.

Multiplying first would give 6^2 = 36, which is 2 \times 3 \times 2 \times 3: two factors of 2, when only one was written. The exponent sits on the 3 alone, so only the 3 is repeated. Doing powers first keeps 3^2 an honest short form of 3 \times 3: swap one for the other anywhere in an expression and the value never changes.

The same reason puts multiplying before adding. 6 + 2 \times 5 is short for 6 + 5 + 5, which is 16. Adding first, 8 \times 5 = 40, would repeat the 6 as well. A power is short for repeated multiplying, and multiplying is short for repeated adding, so each short form is worked out before the level it stands for.

Why left to right on one level

Reading left to right on one level is a convention, the agreed way to read the line, like reading a sentence. What it protects is that each sign belongs to the number just after it. In 24 \div 4 \times 2 the 24 is divided by 4 and multiplied by 2.

Dividing by a number is the same as multiplying by its reciprocal, as in Dividing fractions, so the line can be rewritten with no division sign left in it:

24 \div 4 \times 2 = 24 \times \tfrac{1}{4} \times 2 = 12

Written as a product, its factors can be multiplied in any order and the value stays 12. Reading the original line left to right gives the same: 24 \div 4 = 6, then 6 \times 2 = 12. Working 4 \times 2 first would give 24 \div 8 = 3, which divides by the 2 that the expression multiplies by. That is why multiplying and dividing can share a level: read left to right, each sign stays with its own number.

Adding and subtracting share a level for the same reason. In 20 - 6 + 4 the minus sign belongs to the 6 alone: take 6 away, then put 4 back, which leaves 18. Adding 6 + 4 first would take the 4 away as well and give 10.

Grouping symbols come first

Grouping symbols exist to override the order: whatever they hold is worked out first and then treated as one number. When brackets sit inside brackets, the innermost pair comes first. The outer pair is often written square, [\;], only so the pairs are easy to match.

A fraction bar groups too. Its top is worked out completely, and so is its bottom, and only then is the top divided by the bottom, which is what a fraction means. An exponent groups whatever is raised: 2^{1 + 2} = 2^3 = 8.

A power of a bracket, and a power inside one

An exponent reaches only its base: the single number, or the whole bracket, written just before it. So (2 + 3)^2 squares the sum, 5^2 = 25, while 2 + 3^2 squares the 3 alone and adds 2, which is 11.

Figure 1The bracket makes the whole sum the base, a square of side 5. Without it only the 3 is squared, and the 2 is added on afterwards.

The square of side 5 holds the 3 by 3 square, but also a 2 by 2 square and two 2 by 3 strips, and none of those are in 2 + 3^2.

Worked examples

Example 1

Powers before multiplying and dividing

Evaluate (a) 5 + 4 \times 2^3 and (b) 6^2 \div 4 \times 3.

Solution

(a) There are no grouping symbols. The power comes first, 2^3 = 8, then the product, then the sum:

5 + 4 \times 2^3 = 5 + 4 \times 8 = 5 + 32 = 37

The exponent is written on the 2 alone, so the 4 is not cubed.

(b) The power first, 6^2 = 36. Dividing and multiplying share a level, so work left to right:

6^2 \div 4 \times 3 = 36 \div 4 \times 3 = 9 \times 3 = 27

Example 2

A power of a sum against a sum of powers

Evaluate (a) (4 + 6)^2, (b) 4 + 6^2 and (c) 4^2 + 6^2.

Solution

(a) The bracket comes first: 4 + 6 = 10, and 10^2 = 100.

(b) No bracket, so the exponent reaches only the 6: 4 + 36 = 40.

All three use the same numbers and give three different values. Squaring a sum is not the same as adding the squares: as in Figure 1, the square of side 10 cut into 4 and 6 holds 16 + 36 and also two 4 by 6 strips, and 52 + 24 + 24 = 100.

Example 3

A fraction bar as grouping

Evaluate \dfrac{3^3 - 7}{2^2 + 1}, and write it on one line.

Solution

The bar groups the top and the bottom. Top: 3^3 - 7 = 27 - 7 = 20. Bottom: 2^2 + 1 = 4 + 1 = 5. Then divide: 20 \div 5 = 4.

On one line the grouping has to be written as brackets, (3^3 - 7) \div (2^2 + 1). Leaving them out gives 3^3 - 7 \div 2^2 + 1, which divides only the 7:

27 - \tfrac{7}{4} + 1 = 26\tfrac{1}{4}

Example 4

Brackets inside brackets

Evaluate \left[2 + (7 - 4)^2\right]^2 \div 11.

Solution

Start with the innermost bracket, 7 - 4 = 3. Still inside the square bracket, the power comes before the sum:

\begin{aligned} \left[2 + (7 - 4)^2\right]^2 \div 11 &= \left[2 + 3^2\right]^2 \div 11 \\ &= [2 + 9]^2 \div 11 \\ &= 11^2 \div 11 \\ &= 121 \div 11 = 11 \end{aligned}

The outer exponent waited until the whole square bracket was one number, 11.

Example 5

Fractions and decimals

Evaluate (a) \left(\tfrac{1}{2}\right)^3 \times 16 + 0.4^2 and (b) \tfrac{2}{3} \times \left(1 + \tfrac{1}{2}\right)^2.

Solution

(a) Two powers first: \left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8} and 0.4^2 = 0.16, since tenths times tenths are hundredths. Then multiply, \tfrac{1}{8} \times 16 = 2, and add: 2 + 0.16 = 2.16.

(b) The bracket first: 1 + \tfrac{1}{2} = \tfrac{3}{2}. Then the power, \left(\tfrac{3}{2}\right)^2 = \tfrac{9}{4}, then the product, as in Multiplying fractions:

\tfrac{2}{3} \times \tfrac{9}{4} = \tfrac{18}{12} = \tfrac{3}{2}

Example 6

The comet-watching field

Juna’s comet-watching club uses a square field 12 meters on each side. A square 4 meters on each side is fenced off for the telescope. Isak shares the rest into 8 plots of equal area. Write one expression for the area of each plot and evaluate it.

Solution

A square’s area is its side squared, so the open area is 12^2 - 4^2 square meters, and all of it is divided by 8. The division must wait for the subtraction, so the difference goes in brackets:

(12^2 - 4^2) \div 8 = (144 - 16) \div 8 = 128 \div 8 = 16

Each plot is 16 square meters. Without the brackets, 12^2 - 4^2 \div 8 = 144 - 2 = 142, which divides only the telescope’s square into eight.

Example 7

Placing brackets to change the value

With no brackets, 2 + 4^2 \div 2 \times 3 is 26. Place one pair of brackets so that its value is (a) 54 and (b) 27.

Solution

First, why 26: the power, 16; then left to right, 16 \div 2 = 8 and 8 \times 3 = 24; then 2 + 24 = 26.

(a) 54 is 18 \times 3, and 18 is 36 \div 2. A 36 would come from 6^2, so make the base the sum:

(2 + 4)^2 \div 2 \times 3 = 36 \div 2 \times 3 = 54

(b) 27 is 9 \times 3, and 9 is 18 \div 2. An 18 is 2 + 16, so group the sum after the power:

(2 + 4^2) \div 2 \times 3 = 18 \div 2 \times 3 = 27

A pair around 2 \times 3 gives a third value, 2 + 16 \div 6 = 4\tfrac{2}{3}. Brackets choose which of many possible calculations the line describes.

Boundaries

An exponent reaches only its base. That is the single number or whole bracket just before it. A fraction raised to a power needs brackets, \left(\tfrac{1}{2}\right)^3, because \tfrac{1^3}{2} cubes only the top.

Multiplying and dividing are one level, not two. Neither always comes before the other; whichever is further left comes first. The same is true of adding and subtracting.

The order is about reading, not about what is true. When a different order is wanted, brackets say so, and they always win.

A bottom that works out to 0 gives no value. \dfrac{5}{3^2 - 9} has a bottom of 0, and division by 0 has no answer.

Every number here is zero or more. With negative numbers, a minus sign can also mean “the opposite of”, and where a power reaches becomes a question about signs; that is Order of operations with negative numbers.

Where this goes next

The same order is how a formula with a letter is worked out once a number is put in place of the letter, which is Evaluating expressions and formulas. Squares and cubes inside expressions lean on the values listed in Perfect squares and perfect cubes.

Exercises

Powers before the other operations

Evaluate.

3 + 5^2
2 \times 5^3
50 - 8^2 \div 4
7^2 - 3 \times 2^4
10^3 \div 5^2 \times 2

Brackets and powers

(13 - 9)^3
(3 \times 4)^2 and 3 \times 4^2
6 + (12 \div 3)^2
(1 + 2 + 3)^2 - (1^3 + 2^3 + 3^3)

Fraction bars and nested brackets

\dfrac{5^2 + 11}{2^2}
\dfrac{(5 + 1)^2}{5^2 + 1^2 - 8}
2 \times \left[3 + (5 - 1)^2\right]
\left[(2 + 1)^2 - 6\right]^3
200 - 3 \times \left[2^3 - (6 - 4)\right]^2

Fractions and decimals

\left(\tfrac{3}{5}\right)^2 \times 25 - 4
0.5^2 + 0.5 \times 2^2
\left(\tfrac{1}{3} + \tfrac{1}{2}\right)^2 \div \tfrac{5}{12}
(1.5 - 0.7)^2 \times 10

In context

At the bowling alley, Kemal stacks boxes of score cards into two cubes: one 4 boxes long, wide and high, and one 3 boxes long, wide and high. All the boxes are shared equally among 7 lanes. Write one expression for the boxes each lane gets, and evaluate it.
The comet-watching club has 2^5 hand warmers, and Lea brings 3 packs of 2^2 more. They are shared equally among the club’s 11 members. Write one expression for each member’s share, and evaluate it.

Placing brackets

Place one pair of brackets in 5 + 3^2 - 2 \times 2 so that its value is 60.
Find the value of 1 + 2^3 \times 2. Then place one pair of brackets to make it 54, and another pair to make it 18.

Reasoning

Using what a power means, explain why 4 \times 3^2 is 36 and not 144.
Using a square picture like Figure 1, explain why (5 + 2)^2 is more than 5^2 + 2^2, and by how much.
Explain why 36 \div 6 \div 2 is 3 and not 12.

Answers

28. The power first: 3 + 25.
250. 5^3 = 125, and 2 \times 125 = 250.
34. 8^2 = 64, 64 \div 4 = 16, 50 - 16 = 34.
1. 49 - 3 \times 16 = 49 - 48.
80. 1000 \div 25 \times 2; left to right, 40 \times 2.
64. The bracket first: 4^3.
144 and 48. (3 \times 4)^2 = 12^2, while in 3 \times 4^2 only the 4 is squared: 3 \times 16.
22. 12 \div 3 = 4, 4^2 = 16, 6 + 16.
0. 6^2 = 36 and 1 + 8 + 27 = 36.
9. Top 25 + 11 = 36, bottom 4, and 36 \div 4 = 9.
2. Top 6^2 = 36, bottom 25 + 1 - 8 = 18, and 36 \div 18 = 2.
38. Innermost first: 4^2 = 16, 3 + 16 = 19, 2 \times 19.
27. 3^2 = 9, 9 - 6 = 3, 3^3 = 27.
92. 6 - 4 = 2, 8 - 2 = 6, 6^2 = 36, 3 \times 36 = 108, 200 - 108.
5. \tfrac{9}{25} \times 25 = 9, and 9 - 4 = 5.
2.25. 0.5^2 = 0.25 and 0.5 \times 4 = 2, so 0.25 + 2.
\tfrac{5}{3}. \tfrac{1}{3} + \tfrac{1}{2} = \tfrac{5}{6}, squared \tfrac{25}{36}, and \tfrac{25}{36} \times \tfrac{12}{5} = \tfrac{5}{3}.
6.4. 0.8^2 = 0.64, and 0.64 \times 10 = 6.4.
(4^3 + 3^3) \div 7 = 13. 64 + 27 = 91, and 91 \div 7 = 13. The brackets make the division wait for the whole sum.
(2^5 + 3 \times 2^2) \div 11 = 4. 32 + 3 \times 4 = 44, and 44 \div 11 = 4.
(5 + 3)^2 - 2 \times 2 = 60. 8^2 = 64, and 64 - 4 = 60.
17; (1 + 2)^3 \times 2 = 54; (1 + 2^3) \times 2 = 18. Without brackets, 1 + 8 \times 2 = 1 + 16. Then 3^3 \times 2 = 27 \times 2, and 9 \times 2 = 18.
3^2 is short for 3 \times 3, so 4 \times 3^2 = 4 \times 3 \times 3 = 36. The value 144 is (4 \times 3)^2 = 4 \times 3 \times 4 \times 3, which has two factors of 4, but only one 4 was written and the exponent sits on the 3 alone.
(5 + 2)^2 = 49 and 5^2 + 2^2 = 29, so it is 20 more. A square of side 5 + 2 splits into a 5 by 5 square, a 2 by 2 square and two 5 by 2 strips. The strips, 10 each, are the 20 that 5^2 + 2^2 leaves out.
Each division sign belongs to the number after it, so 36 is divided by 6 and also by 2. As a product that is 36 \times \tfrac{1}{6} \times \tfrac{1}{2} = 3, and left to right gives the same: 36 \div 6 = 6, then 6 \div 2 = 3. Working 6 \div 2 first gives 36 \div 3 = 12, which is 36 \div 6 \times 2: it multiplies by the 2 that the expression divides by.