2.1 order of operations w
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If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
A formal record of the account is shown here.
$5-bills $10-bills
Number of Bills $ Value
23
Total:
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
$5-bills $10-bills
Number of Bills $ Value
23
Total:
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total:
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
2(5) + 3(10) = 40 ($)
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40We may also record the calculation simply as:
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
2(5) + 3(10) = 40 ($)
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40
The calculation to be performed “2(5) + 3(10)” is an expression.
We may also record the calculation simply as:
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
A formal record of the account is shown here.
2(5) + 3(10) = 40 ($)
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40
The calculation to be performed “2(5) + 3(10)” is an expression.
We may also record the calculation simply as:
An arithmetical expression is a calculation procedure written using numbers and arithmetic operational symbols.
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
The statement “2(5) + 3(10) = 40” is called an equation, i.e. we are equating or proclaiming that 2(5) + 3(10) = 40.
A formal record of the account is shown here.
2(5) + 3(10) = 40 ($)
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40
The calculation to be performed “2(5) + 3(10)” is an expression.
We may also record the calculation simply as:
An arithmetical expression is a calculation procedure written using numbers and arithmetic operational symbols.
If we have two $5-bills and three $10-bills, we have $40 in total.
Order of Operations
To obtain the correct answer 40, we multiply 2 x 5 = 10 and 3 x 10 = 30, then we add the products 10 and 30.
The statement “2(5) + 3(10) = 40” is called an equation, i.e. we are equating or proclaiming that 2(5) + 3(10) = 40.
A formal record of the account is shown here.
2(5) + 3(10) = 40 ($)
$5-bills $10-bills
Number of Bills $ Value
23
2x5=$10 3x10=$30
Total: $40
The calculation to be performed “2(5) + 3(10)” is an expression.
We may also record the calculation simply as:
An arithmetical expression is a calculation procedure written using numbers and arithmetic operational symbols.
We will study equations more later.
Order of OperationsOrder of Operations
Order of OperationsOrder of Operations Given an arithmetic expression, we perform the operations, from the in the following order (excluding taking powers).
Example A.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
Order of OperationsOrder 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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
Order of OperationsOrder 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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 4 + 3(7)
Order of OperationsOrder 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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 4 + 3(7)
Order of OperationsOrder 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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 4 + 3(7)= 4 + 21
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 4 + 3(7)= 4 + 21
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 4 + 3(7)= 4 + 21
= 25
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 4 + 3(7)= 4 + 21
= 25
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 17
= 4 + 3(7)= 4 + 21
= 25
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 17
= 4 + 3(7)= 4 + 21
= 25
= 9 – 2[11 – 3(3)]
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 17
= 4 + 3(7)= 4 + 21
= 25
= 9 – 2[11 – 3(3)]= 9 – 2[11 – 9]
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 17
= 4 + 3(7)= 4 + 21
= 25
= 9 – 2[11 – 3(3)]= 9 – 2[11 – 9]
= 9 – 2[2]
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]a. 4 + 3(5 + 2) b. 4(8) – 3(5)
= 32 – 15 = 17
= 4 + 3(7)= 4 + 21
= 25
= 9 – 2[11 – 3(3)]= 9 – 2[11 – 9]
= 9 – 2[2]= 9 – 4 = 5
Order of Operations
3rd. Do additions and subtractions (from left to right).
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.c. 9 – 2[11 – 3(2 + 1)]
(Don’t perform “4 + 3” or “9 – 2” in the above problems!!)
a. 4 + 3(5 + 2) b. 4(8) – 3(5) = 32 – 15
= 17= 4 + 3(7)= 4 + 21
= 25
= 9 – 2[11 – 3(3)]= 9 – 2[11 – 9]
= 9 – 2[2]= 9 – 4 = 5
Order of OperationsExponents
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
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.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
In particular 2 is 21, 3 is 31 etc..
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
In particular 2 is 21, 3 is 31 etc..Hence 2 x 32 (= 2*32) is 21 x 32 = 2 x 3 x 3
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
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.
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In other words, to compute the expression 2 x 32 we do the power 32 first.
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
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.
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In other words, to compute the expression 2 x 32 we do the power 32 first.
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
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.
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?
Example B. Write down the arithmetic expressions for computing the following and find their answers.
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In other words, to compute the expression 2 x 32 we do the power 32 first.
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
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.
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?
Example B. Write down the arithmetic expressions for computing the following and find their answers.
There are 4 x 4 = 42 or 16 slices of pizza.
N is called the exponent, or the power of the base x.
Order of Operations
Recall that we write 22 for 2*2, 23 for 2*2*2, 24 for 2*2*2*2 and so on.
Exponents
In other words, to compute the expression 2 x 32 we do the power 32 first.
In general, we write x*x*x…*x as xN where N is the number of x’s multiplied to itself.
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.
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?
Example B. Write down the arithmetic expressions for computing the following and find their answers.
There are 4 x 4 = 42 or 16 slices of pizza.There are 5 x 5 = 52 or 25 pieces of cake.
N is called the exponent, or the power of the base x.
Order of Operationsb. 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?
Order of Operationsb. 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.
Order of Operationsb. 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 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
= 3*16 = $48.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
= 3*16 = $48.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
= 3*16 = $48.
= 48 + 50 = $98.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
c. The total sale is to be shared by 7 people, how much does each person get?
= 3*16 = $48.
= 48 + 50 = $98.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
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
= 3*16 = $48.
= 48 + 50 = $98.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
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
= 3*16 = $48.
= 48 + 50 = $98.
or 98 ÷ 7 = $14.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
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
d. If we make three such batches of the square pizzas and cakes, how much would each person get then?
= 3*16 = $48.
= 48 + 50 = $98.
or 98 ÷ 7 = $14.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
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
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
= 3*16 = $48.
= 48 + 50 = $98.
or 98 ÷ 7 = $14.
Order of Operationsb. 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
Hence the total sale is 3*42 + 2*52 Each piece cost $2 so the cake can make 2*52 = 2*25 = $50.
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
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
= 3*16 = $48.
= 48 + 50 = $98.
or 98 ÷ 7 = $14.
= 98 ÷ 7 x 3 = 14 x 3 = $42.
Order of OperationsHere is the “order of operations” including raising powers.Order of Operations (PEMDAS)
Order of OperationsHere 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.
Order of OperationsHere 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).
Order of OperationsHere 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.
Order of Operations
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
d. (3 + 2)3 c. 33 + 23
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2)
d. (3 + 2)3 c. 33 + 23
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2)
d. (3 + 2)3 c. 33 + 23
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
= (6)(6)(6) = 216
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
= (6)(6)(6) = 216
= 27 + 8
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
= (6)(6)(6) = 216
= 27 + 8 = 35
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
Do the ( ) first, so (3 + 2)3 = (5)3
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
= (6)(6)(6) = 216
= 27 + 8 = 35
Order of Operations
b. (3*2)3 Example C. Calculate. a. 3*23
3*23 (= 3*2*2*2) (3*2)3 = (6)3
d. (3 + 2)3 c. 33 + 23
33 + 23 = 3*3*3 + 2*2*2
Do the ( ) first, so (3 + 2)3 = (5)3
4th. (Addition and Subtraction) Do additions and subtractions in order, from left to right.
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.
= 3*8 = 24Do the ( ) first,
= (6)(6)(6) = 216
= 27 + 8 = 35 = (5)(5)(5) = 125
Order of Operationse. 24÷3 x 22
Order of Operationse. 24÷3 x 22
= 24÷3 x 4
Order of Operationse. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
= 2{23 + [24 – 32(2)] }
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
= 2{23 + [24 – 32(2)] }= 2{23 + [24 – 9(2)] }
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
= 2{23 + [24 – 32(2)] }= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
= 2{23 + [24 – 32(2)] }= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }
= 2{8 + 6}
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
Order of Operations
f. 2{23 + [24 – 32(8 – 6)] }
For a lengthy problem, perform the operations vertically so each step can be tracked easily.
= 2{23 + [24 – 32(2)] }= 2{23 + [24 – 9(2)] }
= 2{23 + [24 –18] }
= 2{8 + 6}
= 2{14} = 28
e. 24÷3 x 22
= 24÷3 x 4= 8 x 4 = 32
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.
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