8 multiplication division of signed numbers, order of operations

Multiplication and Division of Signed Numbers

Rule for Multiplication of Signed NumbersMultiplication and Division of Signed Numbers

Rule for Multiplication of Signed NumbersTo multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product.


Rule for Multiplication of Signed NumbersTo multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ;


Rule for Multiplication of Signed NumbersTo multiple two signed numbers, we multiply their absolute values and use the following rules for the sign of the product. + * + = – * – = + ; + * – = – * + = – ;




Two numbers with the same sign multiplied yield positive products.



Two numbers with the same sign multiplied yield a positive product. Two numbers with opposite signs multiplied yield a negative product.




Example A.

a. 5 * (4) = –5 * (–4)




Example A.

a. 5 * (4) = –5 * (–4) = 20




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4)




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20In algebra, multiplication operation are not always written down explicitly.




Example A.

a. 5 * (4) = –5 * (–4) = 20 b. –5 * (4) = 5 * (–4) = –20In algebra, multiplication operation are not always written down explicitly. Instead we use the following rules to identify multiplication operations.

Multiplication and Division of Signed Numbers● If there is no operation indicated between two quantities, the operation between them is multiplication.

Multiplication and Division of Signed Numbers● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y.


● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication.

● If there is no operation indicated between two quantities, the operation between them is multiplication. Hence xy means x * y.


● If there is no operation indicated between a set of ( ) and a quantity, the operation between them is multiplication. Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.



● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication.




● If there is no operation indicated between two sets of ( )’s, the operation between them is multiplication. Hence (x + y)(a + b) = (x + y) * (a + b)







However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine.





However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15,





However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,





However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25,





However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.


However, if there is a “+” or “–” sign between the ( ) and a quantity, then the operation is to combine. Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8, and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.



To multiply many signed numbers together, we always determine the sign of the product first, then multiply just the numbers themselves. The sign of the product is determined by the following Even–Odd Rules.


Even-Odd Rule for the Sign of a Product• If there are even number of negative numbers in the multiplication, the product is positive.


Even-Odd Rule for the Sign of a Product• If there are even number of negative numbers in the multiplication, the product is positive.• If there are odd number of negative numbers in the multiplication, the product is negative.



Example B.

a. –1(–2 ) 2 (–1)



Example B.

a. –1(–2 ) 2 (–1)


three negative numbers, so the product is negative


Example B.

a. –1(–2 ) 2 (–1) = – 4



4 came from 1*2*2*1 (just the numbers)


Example B.

a. –1(–2 ) 2 (–1) = – 4

b. (–2)4




Example B.

a. –1(–2 ) 2 (–1) = – 4

b. (–2)4 = (–2 )(–2)(–2)(–2)




Example B.

a. –1(–2 ) 2 (–1) = – 4

b. (–2)4 = (–2 )(–2)(–2)(–2)



four negative numbers, so the product is positive


Example B.

a. –1(–2 ) 2 (–1) = – 4

b. (–2)4 = (–2 )(–2)(–2)(–2) = 16





Example B.

a. –1(–2 ) 2 (–1) = – 4

b. (–2)4 = (–2 )(–2)(–2)(–2) = 16

Fact: A quantity raised to an even power is always positive i.e. xeven is always positive (except 0).




Multiplication and Division of Signed NumbersIn algebra, a ÷ b is written as a/b or . a

b

Rule for the Sign of a Quotient


b

Rule for the Sign of a QuotientDivision of signed numbers follows the same sign-rules for multiplications.


b



b

++ = –

– = + ++ = – –=–


Two numbers with the same sign divided yield a positive quotient.


b

++ = –

– = + ++ = – –=–


Two numbers with the same sign divided yield a positive quotient. Two numbers with opposite signs divided yield a negative quotient.


b

++ = –

– = + ++ = – –=–




Example C.

a.

In algebra, a ÷ b is written as a/b or . ab

++ = –

– = + ++ = – –=–

204 = –20

–4




Example C.

a.


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4)


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=–4




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=36–4–4




Example C.

a.

b . –20 / 4 = 20 / (–4) = –5


++ = –

– = + ++ = – –=–

204 = –20

–4 = 5

c. (–6)2

=36–4 =–4 –9

The Even–Odd Rule applies to more length * and / operations problems.




Example D. Simplify.(– 4)6(–1)(–3)(–2)(–5)12




five negative numbersso the product is negative




= –





= –


simplify just the numbers 4(6)(3)2(5)(12)




= –



= – 35




= –



= – 35Various form of the Even–Odd Rule extend to algebra and geometry. It’s the basis of many decisions and conclusions in mathematics problems. The following is an example of the two types of graphs there are due to this Even–Odd Rule. (Don’t worry about how they are produced.)

Order of Operations

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)

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)

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)]

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)]

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.

The Even Power Graphs vs. Odd Power Graphs of y = xN


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

1. 3 – 3 2. 3(–3) 3. (3) – 3 4. (–3) – 3

5. –3(–3) 6. –(–3)(–3) 7. (–3) – (–3) 8. –(–3) – (–3)B. Multiply. Determine the sign first.9. 2(–3) 10. (–2)(–3) 11. (–1)(–2)(–3) 12. 2(–2)(–3) 13. (–2)(–2)(–2) 14. (–2)(–2)(–2)(–2) 15. (–1)(–2)(–2)(–2)(–2) 16. 2(–1)(3)(–1)(–2)

17. 12–3 18. –12

–3 19. –24–8

21. (2)(–6)–8

C. Simplify. Determine the sign and cancel first.

20. 24–12

22. (–18)(–6)–9

23. (–9)(6)(12)(–3)

24. (15)(–4)(–8)(–10)

25. (–12)(–9)(– 27)(15)

26. (–2)(–6)(–1) (2)(–3)(–2)

27. 3(–5)(–4)(–2)(–1)(–2)

28. (–2)(3)(–4)5(–6)(–3)(4)(–5)6(–7)


Education

8 multiplication division of signed numbers, order of operations