Order of Operations, Step by Step

Jump to section:

Key Ideas + − × ÷ Parenthesis and Exponents Nested Parenthesis Negative Numbers Absolute Value Quiz

What to Remember

The order

Parentheses first. Then exponents. Then multiplication and division from left to right.

Addition and subtraction come last, from left to right.

Multiplication does not come before division. Do whichever comes first as you read.

Nested parentheses

When one pair sits inside another, do the innermost pair first.

Each pair is written with parentheses ( ).

The exponent stays with its base

3² means 3 × 3.

−4² means the negative of 4². The base is 4.

(−4)² means the base is −4.

Absolute value bars

Work inside the bars first. Then drop the sign.

The bars group the inside the way parentheses do.

Easy 1

+ − × ÷

Multiply and divide from left to right. Then add and subtract from left to right.

Example

Simplify.

6 + 15 ÷ 3 × 2 − 4

6	+	15	÷	3	×	2	−	4
6	+	5			×	2	−	4
6	+	10					−	4
16							−	4

Simplify.

12 + 16 ÷ 4 × 6 − 5

12	+	16	÷	4	×	6	−	5
12	+	4			×	6	−	5
12	+	24					−	5
36							−	5

Simplify.

8 + 20 ÷ 5 × 3 − 4

8	+	20	÷	5	×	3	−	4
8	+	4			×	3	−	4
8	+	12					−	4
20							−	4

Simplify.

6 × 7 + 18 ÷ 3 + 4

6	×	7	+	18	÷	3	+	4
42			+	18	÷	3	+	4
42			+	6			+	4
48							+	4

Simplify.

4 × 9 + 24 ÷ 6 + 5

4	×	9	+	24	÷	6	+	5
36			+	24	÷	6	+	5
36			+	4			+	5
40							+	5

Simplify.

7 × 5 + 8 − 6 + 11

7	×	5	+	8	−	6	+	11
35			+	8	−	6	+	11
43					−	6	+	11
37							+	11

Simplify.

9 × 3 + 12 − 7 + 6

9	×	3	+	12	−	7	+	6
27			+	12	−	7	+	6
39					−	7	+	6
32							+	6

Simplify.

8 × 8 − 5 × 9 + 6

8	×	8	−	5	×	9	+	6
64			−	5	×	9	+	6
64			−	45			+	6
19							+	6

Simplify.

7 × 6 − 4 × 8 + 9

7	×	6	−	4	×	8	+	9
42			−	4	×	8	+	9
42			−	32			+	9
10							+	9

Simplify.

9 − 4 + 3 × 5 × 2

9	−	4	+	3	×	5	×	2
9	−	4	+	15			×	2
9	−	4	+	30
5			+	30

Simplify.

12 − 5 + 2 × 6 × 4

12	−	5	+	2	×	6	×	4
12	−	5	+	12			×	4
12	−	5	+	48
7			+	48

Easy 2

Parenthesis and Exponents

Parentheses first. Then exponents.

Example

Simplify.

(5 + 1)² + (10 − 16 ÷ 8)

(	5	+	1	)	²	+	(	10	−	16	÷	8	)
6					²	+	(	10	−	16	÷	8	)
6					²	+	(	10	−	2			)
6					²	+	8
36						+	8

Simplify.

(8 + 22 − 6) ÷ 4 − 2²

(	8	+	22	−	6	)	÷	4	−	2	²
(	30			−	6	)	÷	4	−	2	²
24							÷	4	−	2	²
24							÷	4	−	4
6									−	4

Simplify.

(7 + 15 − 4) ÷ 6 − 3²

(	7	+	15	−	4	)	÷	6	−	3	²
(	22			−	4	)	÷	6	−	3	²
18							÷	6	−	3	²
18							÷	6	−	9
3									−	9

Simplify.

(9 + 27 − 6) ÷ 5 + 4²

(	9	+	27	−	6	)	÷	5	+	4	²
(	36			−	6	)	÷	5	+	4	²
30							÷	5	+	4	²
30							÷	5	+	16
6									+	16

Simplify.

(12 + 18 − 6) ÷ 8 + 2²

(	12	+	18	−	6	)	÷	8	+	2	²
(	30			−	6	)	÷	8	+	2	²
24							÷	8	+	2	²
24							÷	8	+	4
3									+	4

Simplify.

3 × (4 × 5 + 2²) + 6

3	×	(	4	×	5	+	2	²	)	+	6
3	×	(	4	×	5	+	4		)	+	6
3	×	(	20			+	4		)	+	6
3	×	24								+	6
72										+	6

Simplify.

2 × (6 × 3 + 4²) + 5

2	×	(	6	×	3	+	4	²	)	+	5
2	×	(	6	×	3	+	16		)	+	5
2	×	(	18			+	16		)	+	5
2	×	34								+	5
68										+	5

Simplify.

(5 + 2)² + (11 − 12 ÷ 3)

(	5	+	2	)	²	+	(	11	−	12	÷	3	)
7					²	+	(	11	−	12	÷	3	)
7					²	+	(	11	−	4			)
7					²	+	7
49						+	7

Simplify.

(4 + 4)² + (15 − 18 ÷ 6)

(	4	+	4	)	²	+	(	15	−	18	÷	6	)
8					²	+	(	15	−	18	÷	6	)
8					²	+	(	15	−	3			)
8					²	+	12
64						+	12

Simplify.

(8 − 3)² − (4 + 10 ÷ 5)

(	8	−	3	)	²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	2			)
5					²	−	6
25						−	6

Simplify.

5 × (3 × 2 + 3²) − 7

5	×	(	3	×	2	+	3	²	)	−	7
5	×	(	3	×	2	+	9		)	−	7
5	×	(	6			+	9		)	−	7
5	×	15								−	7
75										−	7

Medium

Nested Parenthesis

Start with the innermost parentheses.

Example

Simplify.

(10 + (12 ÷ 3)) + 7

(	10	+	(	12	÷	3	)	)	+	7
(	10	+	4					)	+	7
14									+	7

Simplify.

(16 + (18 ÷ 6)) + 5

(	16	+	(	18	÷	6	)	)	+	5
(	16	+	3					)	+	5
19									+	5

Simplify.

14 + ((9 + 6) + 3)

14	+	(	(	9	+	6	)	+	3	)
14	+	(	15					+	3	)
14	+	18

Simplify.

((12 + 6) × 4) − 8

(	(	12	+	6	)	×	4	)	−	8
(	18					×	4	)	−	8
72									−	8

Simplify.

7 + ((11 + 5) × 3)

7	+	(	(	11	+	5	)	×	3	)
7	+	(	16					×	3	)
7	+	48

Simplify.

(15 + (20 ÷ 4 + 6))

(	15	+	(	20	÷	4	+	6	)	)
(	15	+	(	5			+	6	)	)
(	15	+	11							)

Simplify.

(11 + (18 ÷ 3 − 2))

(	11	+	(	18	÷	3	−	2	)	)
(	11	+	(	6			−	2	)	)
(	11	+	4							)

Simplify.

13 + (5 × (8 + 4))

13	+	(	5	×	(	8	+	4	)	)
13	+	(	5	×	12					)
13	+	60

Simplify.

9 + (4 + (17 − 6))

9	+	(	4	+	(	17	−	6	)	)
9	+	(	4	+	11					)
9	+	15

Simplify.

(20 + (24 ÷ 6)) + 3

(	20	+	(	24	÷	6	)	)	+	3
(	20	+	4					)	+	3
24									+	3

Simplify.

((14 + 9) + 6) − 8

(	(	14	+	9	)	+	6	)	−	8
(	23					+	6	)	−	8
29									−	8

Hard

Negative Numbers

Problems like + − × ÷, Parenthesis and Exponents, and Nested Parenthesis, with negative numbers.

Example

Simplify.

−3² + 4 × (2 − (−1))

−	3	²	+	4	×	(	2	−	(−1)	)
−	3	²	+	4	×	3
−	9		+	4	×	3
−9			+	4	×	3
−9			+	12

Simplify.

6 + (−18) ÷ 3 × 2 − 4

6	+	(−18)	÷	3	×	2	−	4
6	+	(−6)			×	2	−	4
6	+	(−12)					−	4
−6							−	4

Simplify.

(−4) × 6 − 5 × (−3) + 2

(−4)	×	6	−	5	×	(−3)	+	2
−24			−	5	×	(−3)	+	2
−24			−	(−15)			+	2
−9							+	2

Simplify.

8 − (−5) + (−3) × 4 × 2

8	−	(−5)	+	(−3)	×	4	×	2
8	−	(−5)	+	(−12)			×	2
8	−	(−5)	+	(−24)
13			+	(−24)

Simplify.

(12 + (−8) − 2) ÷ 2 − 3²

(	12	+	(−8)	−	2	)	÷	2	−	3	²
(	4			−	2	)	÷	2	−	3	²
2							÷	2	−	3	²
2							÷	2	−	9
1									−	9

Simplify.

(−5)² + (6 − 15 ÷ (−3))

(−5)	²	+	(	6	−	15	÷	(−3)	)
(−5)	²	+	(	6	−	(−5)			)
(−5)	²	+	11
25		+	11

Simplify.

−4² + (9 − 2) × 3

−	4	²	+	(	9	−	2	)	×	3
−	4	²	+	7					×	3
−	16		+	7					×	3
−16			+	7					×	3
−16			+	21

Simplify.

(10 + ((−12) ÷ 4)) + 6

(	10	+	(	(−12)	÷	4	)	)	+	6
(	10	+	(−3)					)	+	6
7									+	6

Simplify.

((−3 + 8) × 4) − 5

(	(	−3	+	8	)	×	4	)	−	5
(	5					×	4	)	−	5
20									−	5

Simplify.

7 + (2 × (−6 + 10))

7	+	(	2	×	(	−6	+	10	)	)
7	+	(	2	×	4					)
7	+	8

Simplify.

(4 + ((−18) ÷ 3 + 5))

(	4	+	(	(−18)	÷	3	+	5	)	)
(	4	+	(	−6			+	5	)	)
(	4	+	(−1)							)

Challenge 1

Absolute Value

Work inside the bars first. Then drop the sign.

Example

Simplify.

|−5| + 2 × (−3)

\|	−5	\|	+	2	×	(−3)
5			+	2	×	(−3)
5			+	(−6)

Simplify.

|−8 + 3| × 2

\|	−8	+	3	\|	×	2
\|	−5			\|	×	2
5					×	2

Simplify.

4 × (|−6| − 10)

4	×	(	\|	−6	\|	−	10	)
4	×	(	6			−	10	)
4	×	(−4)

Simplify.

−|12 − 5| + 3²

−	\|	12	−	5	\|	+	3	²
−	\|	7			\|	+	3	²
−	7					+	3	²
−7						+	3	²
−7						+	9

Simplify.

|−4| × |−5| − 6

\|	−4	\|	×	\|	−5	\|	−	6
4			×	\|	−5	\|	−	6
4			×	5			−	6
20							−	6

Simplify.

(−3) × |2 − 9| + 15

(−3)	×	\|	2	−	9	\|	+	15
(−3)	×	\|	−7			\|	+	15
(−3)	×	7					+	15
−21							+	15

Simplify.

|−15| ÷ 3 + 2 × (−4)

\|	−15	\|	÷	3	+	2	×	(−4)
15			÷	3	+	2	×	(−4)
5					+	2	×	(−4)
5					+	(−8)

Simplify.

−|−7 − 2| + 4 × 3

−	\|	−7	−	2	\|	+	4	×	3
−	\|	−9			\|	+	4	×	3
−	9					+	4	×	3
−9						+	4	×	3
−9						+	12

Quiz

Ten problems from + − × ÷, Parenthesis and Exponents, and Nested Parenthesis. The last step names the section to open for more help.

Simplify.

5 + 12 ÷ 4 × 3 − 2

5	+	12	÷	4	×	3	−	2
5	+	3			×	3	−	2
5	+	9					−	2
14							−	2

Simplify.

(4 + 5)² + (7 − 8 ÷ 2)

(	4	+	5	)	²	+	(	7	−	8	÷	2	)
9					²	+	(	7	−	8	÷	2	)
9					²	+	(	7	−	4			)
9					²	+	3
81						+	3

Simplify.

(6 + (15 ÷ 3)) + 4

(	6	+	(	15	÷	3	)	)	+	4
(	6	+	5					)	+	4
11									+	4

Simplify.

5 × 6 − 3 × 4 + 7

5	×	6	−	3	×	4	+	7
30			−	3	×	4	+	7
30			−	12			+	7
18							+	7

Simplify.

8 + (3 × (5 + 2))

8	+	(	3	×	(	5	+	2	)	)
8	+	(	3	×	7					)
8	+	21

Simplify.

(6 + 14 − 2) ÷ 9 + 3²

(	6	+	14	−	2	)	÷	9	+	3	²
(	20			−	2	)	÷	9	+	3	²
18							÷	9	+	3	²
18							÷	9	+	9
2									+	9

Simplify.

10 − 3 + 2 × 4 × 3

10	−	3	+	2	×	4	×	3
10	−	3	+	8			×	3
10	−	3	+	24
7			+	24

Simplify.

((7 + 5) × 3) − 6

(	(	7	+	5	)	×	3	)	−	6
(	12					×	3	)	−	6
36									−	6

Simplify.

4 × (2 × 3 + 2²) + 1

4	×	(	2	×	3	+	2	²	)	+	1
4	×	(	2	×	3	+	4		)	+	1
4	×	(	6			+	4		)	+	1
4	×	10								+	1
40										+	1

Simplify.

(9 + (16 ÷ 4 − 1))

(	9	+	(	16	÷	4	−	1	)	)
(	9	+	(	4			−	1	)	)
(	9	+	3							)

Need help with the order of operations?

Book a Session with Amy

(	8	+	22	−	6	)	÷	4	−	2	²
(	30			−	6	)	÷	4	−	2	²
24							÷	4	−	2	²
24							÷	4	−	4
6									−	4

(	7	+	15	−	4	)	÷	6	−	3	²
(	22			−	4	)	÷	6	−	3	²
18							÷	6	−	3	²
18							÷	6	−	9
3									−	9

(	8	−	3	)	²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	2			)
5					²	−	6
25						−	6

(	8	+	22	−	6	)	÷	4	−	2	²
(	30			−	6	)	÷	4	−	2	²
24							÷	4	−	2	²
24							÷	4	−	4
6									−	4

(	7	+	15	−	4	)	÷	6	−	3	²
(	22			−	4	)	÷	6	−	3	²
18							÷	6	−	3	²
18							÷	6	−	9
3									−	9

(	8	−	3	)	²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	2			)
5					²	−	6
25						−	6

(	8	+	22	−	6	)	÷	4	−	2	²
(	30			−	6	)	÷	4	−	2	²
24							÷	4	−	2	²
24							÷	4	−	4
6									−	4

(	7	+	15	−	4	)	÷	6	−	3	²
(	22			−	4	)	÷	6	−	3	²
18							÷	6	−	3	²
18							÷	6	−	9
3									−	9

(	8	−	3	)	²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	10	÷	5	)
5					²	−	(	4	+	2			)
5					²	−	6
25						−	6