9 order of operations

Order of Operations

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If we have two $5-bill and two $10-bills, Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars.

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first,

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first, then we add the products 10 and 20.

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first, then we add the products 10 and 20. If I have three $10-bills and you have four $10-bills,

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first, then we add the products 10 and 20. If I have three $10-bills and you have four $10-bills, we have 3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first, then we add the products 10 and 20. If I have three $10-bills and you have four $10-bills, we have 3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $. In this case, we group the 3 + 4 in the “( )” to indicate that we are to add them first,

Order of Operations

If we have two $5-bill and two $10-bills, we have the total of 2(5) + 2(10) = 30 dollars. To get the correct answer 30, we multiply the 2 and the 5 and multiply the 2 and the10 first, then we add the products 10 and 20. If I have three $10-bills and you have four $10-bills, we have 3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $. In this case, we group the 3 + 4 in the “( )” to indicate that we are to add them first, then multiply the sum to 10.

Order of Operations


Order of Operations

This motivates us to set the rules for the order of operations.


Order of Operations

Order of Operations (excluding raising power)This motivates us to set the rules for the order of operations.


Order of Operations

Order of Operations (excluding raising power)Given an arithmetic expression, we perform the operations in the following order .



Order of Operations

Order of Operations (excluding raising power)Given an arithmetic expression, we perform the operations in the following order . 1st . Do the operations within grouping symbols, starting with the innermost grouping symbol.



Order of Operations

Order of Operations (excluding raising power)Given an arithmetic expression, we perform the operations in the following order . 1st . Do the operations within grouping symbols, starting with the innermost grouping symbol.2nd. Do multiplications and divisions (from left to right).



Order of Operations

Order of Operations (excluding raising power)Given an arithmetic expression, we perform the operations in the following order . 1st . Do the operations within grouping symbols, starting with the innermost grouping symbol.2nd. Do multiplications and divisions (from left to right).3rd. Do additions and subtractions (from left to right).


Example A.

a. 4(–8) + 3(5)

Order of Operations

Example A.

a. 4(–8) + 3(5) = –32 + 15

Order of Operations

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

Order of Operations

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

Order of Operations

b. 4 + 3(5 + 2)

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7)

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)]

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)]

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)] = 9 – 2[7 – 21]

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)] = 9 – 2[7 – 21] = 9 – 2[ –14 ]

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)] = 9 – 2[7 – 21] = 9 – 2[ –14 ] = 9 + 28

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)] = 9 – 2[7 – 21] = 9 – 2[ –14 ] = 9 + 28 = 37

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Example A.

a. 4(–8) + 3(5) = –32 + 15 = –17

c. 9 – 2[7 – 3(6 + 1)] = 9 – 2[7 – 3(7)] = 9 – 2[7 – 21] = 9 – 2[ –14 ] = 9 + 28 = 37

(Don’t perform “4 + 3” or “9 – 2” in the above problems!!)

Order of Operations

b. 4 + 3(5 + 2) = 4 + 3(7) = 4 + 21 = 25

Exercise: Don’t do the part that you shouldn’t do!

1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself.


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base.


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent,


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent, hence 2b3 means 2*b3


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent, hence 2b3 means 2*b3 = 2*b*b*b.


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent, hence 2b3 means 2*b3 = 2*b*b*b. If we want multiply 2b to itself three times, i.e. 2b to the third power,


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent, hence 2b3 means 2*b3 = 2*b*b*b. If we want multiply 2b to itself three times, i.e. 2b to the third power, we write it as (2b)3


1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)

3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]

Order of Operations

Ans: a. 18 b. 18 c. 23 4. 15

ExponentsWe write x*x*x…*x as xN where N is the number of copies of x’s multiplied to itself. N is called the exponent, or the power of x, and x is called the base. The base is the quantity immediately beneath the exponent, hence 2b3 means 2*b3 = 2*b*b*b. If we want multiply 2b to itself three times, i.e. 2b to the third power, we write it as (2b)3 which is (2b)*(2b)*(2b) =8b3.

Order of OperationsExample B. (Exponential Notation)

a. Expand (–3)2 and simplify the answer.


a. Expand (–3)2 and simplify the answer. The base is (–3).


a. Expand (–3)2 and simplify the answer. The base is (–3). Hence (–3)2 is (–3)(–3)


a. Expand (–3)2 and simplify the answer. The base is (–3). Hence (–3)2 is (–3)(–3) = 9

Order of Operations

b. Expand – 32

Example B. (Exponential Notation)


Order of Operations

b. Expand – 32

The base of the 2nd power is 3.



Order of Operations

b. Expand – 32

The base of the 2nd power is 3. Hence – 32 means – (3*3)



Order of Operations

b. Expand – 32

The base of the 2nd power is 3. Hence – 32 means – (3*3) = – 9



c. Expand (3*2)2 and simplify the answer.

Order of Operations

b. Expand – 32




c. Expand (3*2)2 and simplify the answer. The base for the 2nd power is (3*2).

Order of Operations

b. Expand – 32




c. Expand (3*2)2 and simplify the answer. The base for the 2nd power is (3*2). Hence(3*2)2 is (3*2)(3*2)

Order of Operations

b. Expand – 32




c. Expand (3*2)2 and simplify the answer. The base for the 2nd power is (3*2). Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36

Order of Operations

b. Expand – 32





d. Expand 3*22 and simplify the answer.

Order of Operations

b. Expand – 32





d. Expand 3*22 and simplify the answer. The base for the 2nd power is 2.

Order of Operations

b. Expand – 32





d. Expand 3*22 and simplify the answer. The base for the 2nd power is 2. Hence 3*22 means 3*2*2

Order of Operations

b. Expand – 32





d. Expand 3*22 and simplify the answer. The base for the 2nd power is 2. Hence 3*22 means 3*2*2 = 12

Order of Operations

b. Expand – 32




Order of Operationse. Expand (–3y)3 and simplify the answer.

Order of Operationse. Expand (–3y)3 and simplify the answer. (–3y)3 = (–3y)(–3y)(–3y)

Order of Operationse. Expand (–3y)3 and simplify the answer. (–3y)3 = (–3y)(–3y)(–3y) (the product of three negatives number is negative)


= –(3)(3)(3)(y)(y)(y)


= –(3)(3)(3)(y)(y)(y) = –27y3


= –(3)(3)(3)(y)(y)(y) = –27y3

From part b above, we see that the power is to be carried out before multiplication.


= –(3)(3)(3)(y)(y)(y) = –27y3

From part b above, we see that the power is to be carried out before multiplication. Below is the complete rules of order of operations.


= –(3)(3)(3)(y)(y)(y) = –27y3

Order of Operations (PEMDAS)



= –(3)(3)(3)(y)(y)(y) = –27y3

Order of Operations (PEMDAS) 1st. (Parenthesis) Do the operations within grouping symbols, starting with the innermost one.



= –(3)(3)(3)(y)(y)(y) = –27y3

Order of Operations (PEMDAS) 1st. (Parenthesis) Do the operations within grouping symbols, starting with the innermost one.2nd. (Exponents) Do the exponentiation



= –(3)(3)(3)(y)(y)(y) = –27y3

Order of Operations (PEMDAS) 1st. (Parenthesis) Do the operations within grouping symbols, starting with the innermost one.2nd. (Exponents) Do the exponentiation3rd. (Multiplication and Division) Do multiplications and divisions in order from left to right.



= –(3)(3)(3)(y)(y)(y) = –27y3

Order of Operations (PEMDAS) 1st. (Parenthesis) Do the operations within grouping symbols, starting with the innermost one.2nd. (Exponents) Do the exponentiation3rd. (Multiplication and Division) Do multiplications and divisions in order from left to right.4th. (Addition and Subtraction) Do additions and subtractions in order from left to right.


Example C. Order of Operations

a. 52 – 32

Order of Operations


a. 52 – 32 = 25 – 9

Order of Operations


a. 52 – 32 = 25 – 9 = 16

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

d. –32 – 5(3 – 6)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

d. –32 – 5(3 – 6)2

= –9 – 5(–3)2

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

d. –32 – 5(3 – 6)2

= –9 – 5(–3)2

= –9 – 5(9)

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

d. –32 – 5(3 – 6)2

= –9 – 5(–3)2

= –9 – 5(9) = –9 – 45

Order of Operations


a. 52 – 32 = 25 – 9 = 16

b. – (5 – 3)2 = – (2)2

= – 4

c. –2*32 + (2*3)2 = –2*9 + (6)2 = –18 + 36 = 18

d. –32 – 5(3 – 6)2

= –9 – 5(–3)2

= –9 – 5(9) = –9 – 45 = –54

Order of Operations

Make sure that you interpret the operations correctly.Exercise A. Calculate the following expressions.

Order of Operations

7. 1 + 2(3) 8. 4 – 5(6) 9. 7 – 8(–9)

1. 3(–3) 2. (3) – 3 3. 3 – 3(3) 4. 3(–3) + 3 5. +3(–3)(+3) 6. 3 + (–3)(+3)

B. Make sure that you don’t do the ± too early.

10. 1 + 2(3 – 4) 11. 5 – 6(7 – 8) 12. (4 – 3)2 + 1 13. [1 – 2(3 – 4)] – 2 14. 6 + [5 + 6(7 – 8)](+5)15. 1 + 2[1 – 2(3 + 4)] 16. 5 – 6[5 – 6(7 – 8)]17. 1 – 2[1 – 2(3 – 4)] 18. 5 + 6[5 + 6(7 – 8)]19. (1 + 2)[1 – 2(3 + 4)] 20. (5 – 6)[5 – 6(7 – 8)]

C. Make sure that you apply the powers to the correct bases.23. (–2)2 and –22 24 (–2)3 and –23 25. (–2)4 and –24

26. (–2)5 and –25 27. 2*32 28. (2*3)2

21. 1 – 2(–3)(–4) 22. (–5)(–6) – (–7)(–8)

Order of OperationsD. Make sure that you apply the powers to the correct bases.29. (2)2 – 3(2) + 1 30. 3(–2)2 + 4(–2) – 131. –2(3)2 + 3(3) – 5 32. –3(–1)2 + 4(–1) – 433. 3(–2)3 – 4(–2)2 – 1 34. (2)3 – 3(2)2 + 4(2) – 1

35. 2(–1)3 – 3(–1)2 + 4(–1) – 1 36. –3(–2)3 – 4(–2)2 – 4(–2) – 3

37. (6 + 3)2 38. 62 + 32 39. (–4 + 2)3 40. (–4)3 + (2)3

E. Calculate.

41. 72 – 42 42. (7 + 4)(7 – 4 )43. (– 5)2 – 32 44. (–5 + 3)(–5 – 3 )45. 53 – 33 46. (5 – 3) (52 + 5*3 + 32)47. 43 + 23 48. (4 + 2)(42 – 4*2 + 22)

7 – (–5)5 – 353. 8 – 2

–6 – (–2)54.

49. (3)2 – 4(2)(3) 50. (3)2 – 4(1)(– 4)51. (–3)2 – 4(–2)(3) 52. (–2)2 – 4(–1)(– 4)

(–4) – (–8)(–5) – 355. (–7) – (–2)

(–3) – (–6)56.

9 order of operations

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