The document discusses order of operations and exponents. It explains that when evaluating arithmetic expressions, operations should be performed in a specific order: 1) operations within grouping symbols, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Exponents are defined as the number of times a base is used as a factor, with the exponent written as a superscript. Examples are provided to demonstrate applying order of operations correctly.

1. Order of Operations
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2. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.

3. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
Number
$ Value
of Bills
$5-bills
$10-bills
2
3
Total:

4. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
products 10 and 30.
$10-bills
3
Total:

5. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
Total:

6. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
Total:
$40

7. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
We may also record the
Total:
$40
calculation simply as:
2(5) + 3(10) = 40 ($)

8. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
We may also record the
Total:
$40
calculation simply as:
2(5) + 3(10) = 40 ($)
The calculation to be performed “2(5) + 3(10)” is an expression.

9. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
We may also record the
Total:
$40
calculation simply as:
2(5) + 3(10) = 40 ($)
The calculation to be performed “2(5) + 3(10)” is an expression.
An arithmetical expression is a calculation procedure written
using numbers and arithmetic operational symbols.

10. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
We may also record the
Total:
$40
calculation simply as:
2(5) + 3(10) = 40 ($)
The calculation to be performed “2(5) + 3(10)” is an expression.
An arithmetical expression is a calculation procedure written
using numbers and arithmetic operational symbols.
The statement “2(5) + 3(10) = 40” is called an equation,
i.e. we are equating or proclaiming that 2(5) + 3(10) = 40.

11. Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
A formal record of the account is shown here.
To obtain the correct answer 40,
Number
we multiply 2 x 5 = 10 and
$ Value
of Bills
3 x 10 = 30, then we add the
$5-bills
2
2x5=$10
products 10 and 30.
$10-bills
3 3x10=$30
We may also record the
Total:
$40
calculation simply as:
2(5) + 3(10) = 40 ($)
The calculation to be performed “2(5) + 3(10)” is an expression.
An arithmetical expression is a calculation procedure written
using numbers and arithmetic operational symbols.
The statement “2(5) + 3(10) = 40” is called an equation,
i.e. we are equating or proclaiming that 2(5) + 3(10) = 40.
We will study equations more later.

12. Order of Operations
Order of Operations

13. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
Example A.
a. 4 + 3(5 + 2)
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

14. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
starting with the innermost grouping symbol.
Example A.
a. 4 + 3(5 + 2)
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

15. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
starting with the innermost grouping symbol.
Example A.
a. 4 + 3(5 + 2)
= 4 + 3(7)
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

16. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
starting with the innermost grouping symbol.
2nd. Do multiplications and divisions (from left to right).
Example A.
a. 4 + 3(5 + 2)
= 4 + 3(7)
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

17. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
starting with the innermost grouping symbol.
2nd. Do multiplications and divisions (from left to right).
Example A.
a. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

18. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

19. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
c. 9 – 2[11 – 3(2 + 1)]

20. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
c. 9 – 2[11 – 3(2 + 1)]

21. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]

22. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]
= 9 – 2[11 – 3(3)]

23. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]
= 9 – 2[11 – 3(3)]
= 9 – 2[11 – 9]

24. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]
= 9 – 2[11 – 3(3)]
= 9 – 2[11 – 9]
= 9 – 2[2]

25. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]
= 9 – 2[11 – 3(3)]
= 9 – 2[11 – 9]
= 9 – 2[2]
=9–4
=5

26. Order of Operations
Order of Operations
Given an arithmetic expression, we perform the operations,
from the in the following order (excluding taking powers).
1st. Do the operations within grouping symbols (),[ ], or { },
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 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
b. 4(8) – 3(5)
= 32 – 15
= 17
c. 9 – 2[11 – 3(2 + 1)]
= 9 – 2[11 – 3(3)]
= 9 – 2[11 – 9]
= 9 – 2[2]
=9–4
=5
(Don’t perform “4 + 3” or “9 – 2” in the above problems!!)

27. Exponents
Order of Operations

28. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on.

29. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself.

30. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x.

31. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..

32. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3

33. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3 = 2 x 9 = 18.

34. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3 = 2 x 9 = 18.
In other words, to compute the expression 2 x 32 we do the
power 32 first.

35. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3 = 2 x 9 = 18.
In other words, to compute the expression 2 x 32 we do the
power 32 first.
Example B. Write down the arithmetic expressions for
computing the following and find their answers.
a. We bake a square pan pizza and a square cake in one batch.
The pan pizza is cut into 4 rows and 4 columns and the cake
is cut into 5 rows and 5 columns. How many slices of pizza
and how many pieces of cakes do we have?

36. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3 = 2 x 9 = 18.
In other words, to compute the expression 2 x 32 we do the
power 32 first.
Example B. Write down the arithmetic expressions for
computing the following and find their answers.
a. We bake a square pan pizza and a square cake in one batch.
The pan pizza is cut into 4 rows and 4 columns and the cake
is cut into 5 rows and 5 columns. How many slices of pizza
and how many pieces of cakes do we have?
There are 4 x 4 = 42 or 16 slices of pizza.

37. Order of Operations
Exponents
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2
and so on. In general, we write x*x*x…*x as xN where N is the
number of x’s multiplied to itself. N is called the exponent,
or the power of the base x. In particular 2 is 21, 3 is 31 etc..
Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3 = 2 x 9 = 18.
In other words, to compute the expression 2 x 32 we do the
power 32 first.
Example B. Write down the arithmetic expressions for
computing the following and find their answers.
a. We bake a square pan pizza and a square cake in one batch.
The pan pizza is cut into 4 rows and 4 columns and the cake
is cut into 5 rows and 5 columns. How many slices of pizza
and how many pieces of cakes do we have?
There are 4 x 4 = 42 or 16 slices of pizza.
There are 5 x 5 = 52 or 25 pieces of cake.

38. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?

39. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.

40. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.

41. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52

42. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.

43. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?

44. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?
Divide the total sale by 7, so person gets
(3*42 + 2*52) ÷ 7

45. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?
Divide the total sale by 7, so person gets
(3*42 + 2*52) ÷ 7 or 98 ÷ 7 = $14.

46. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?
Divide the total sale by 7, so person gets
(3*42 + 2*52) ÷ 7 or 98 ÷ 7 = $14.
d. If we make three such batches of the square pizzas and
cakes, how much would each person get then?

47. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?
Divide the total sale by 7, so person gets
(3*42 + 2*52) ÷ 7 or 98 ÷ 7 = $14.
d. If we make three such batches of the square pizzas and
cakes, how much would each person get then?
The complete expression for the share of each person is
[(3*42 + 2*52) ÷ 7] x 3

48. Order of Operations
b. We sell the pizza at $3/slice and the cake at $2/piece.
How much money can we make from one pizza?
How much money can we make from one cake?
How much can we make in total?
Each slice cost $3 so the pizza can make 3*42 = 3*16 = $48.
Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
Hence the total sale is 3*42 + 2*52 = 48 + 50 = $98.
c. The total sale is to be shared by 7 people, how much does
each person get?
Divide the total sale by 7, so person gets
(3*42 + 2*52) ÷ 7 or 98 ÷ 7 = $14.
d. If we make three such batches of the square pizzas and
cakes, how much would each person get then?
The complete expression for the share of each person is
[(3*42 + 2*52) ÷ 7] x 3 = 98 ÷ 7 x 3 = 14 x 3 = $42.

49. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)

50. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.

51. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).

52. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (Multiplication and Division) Do multiplications and
divisions in order, from left to right.

53. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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.

54. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
b. (3*2)3
c. 33 + 23
d. (3 + 2)3

55. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8
c. 33 + 23
b. (3*2)3
d. (3 + 2)3

56. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
b. (3*2)3
d. (3 + 2)3

57. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3
d. (3 + 2)3

58. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3 = (6)(6)(6) = 216
d. (3 + 2)3

59. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
= 27 + 8
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3 = (6)(6)(6) = 216
d. (3 + 2)3

60. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
= 27 + 8 = 35
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3 = (6)(6)(6) = 216
d. (3 + 2)3

61. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
= 27 + 8 = 35
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3 = (6)(6)(6) = 216
d. (3 + 2)3
Do the ( ) first,
so (3 + 2)3 = (5)3

62. Order of Operations
Here is the “order of operations” including raising powers.
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation (powers).
3rd. (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. Calculate.
a. 3*23
3*23 (= 3*2*2*2)
= 3*8 = 24
c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
= 27 + 8 = 35
b. (3*2)3
Do the ( ) first,
(3*2)3 = (6)3 = (6)(6)(6) = 216
d. (3 + 2)3
Do the ( ) first,
so (3 + 2)3 = (5)3 = (5)(5)(5) = 125

63. Order of Operations
e. 24÷3 x 22

64. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4

65. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32

66. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }

67. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }
= 2{23 + [24 – 32(2)] }

68. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }
= 2{23 + [24 – 32(2)] }
= 2{23 + [24 – 9(2)] }

69. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }
= 2{23 + [24 – 32(2)] }
= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }

70. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }
= 2{23 + [24 – 32(2)] }
= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }
= 2{8 + 6}

71. Order of Operations
e. 24÷3 x 22
= 24÷3 x 4
= 8 x 4 = 32
For a lengthy problem, perform the operations vertically so
each step can be tracked easily.
f. 2{23 + [24 – 32(8 – 6)] }
= 2{23 + [24 – 32(2)] }
= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }
= 2{8 + 6}
= 2{14} = 28

72. Order of Operations
Exercise A. Calculate the following expressions.
Make sure that you interpret the operations correctly.
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.
7. 1 + 2(3)
8. 4 – 5(6)
9. 7 – 8(–9)
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)]
21. 1 – 2(–3)(–4)
22. (–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

73. Order of Operations
D.Make sure that you apply the powers to the correct bases.
29. (2)2 – 3(2) + 1
30. 3(–2)2 + 4(–2) – 1
31. –2(3)2 + 3(3) – 5
32. –3(–1)2 + 4(–1) – 4
33. 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
E. Calculate.
37. (6 + 3)2
38. 62 + 32
39. (–4 + 2)3
40. (–4)3 + (2)3
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)
49. (3)2 – 4(2)(3)
50. (3)2 – 4(1)(– 4)
51. (–3)2 – 4(–2)(3)
52. (–2)2 – 4(–1)(– 4)
8–2
54. –6 – (–2)
53. 75––(–5)
3
56. (–7) – (–2)
55. (–4) ––(–8)
(–3) – (–6)
(–5) 3