Properties of operations

Question 1

State the formula for the commutative property of addition (1) and multiplication (2)

Answer 1

(1) a + b = b + a
(2) a · b = b · a

Question 2

State the formula for the associate property of addition (1) and multiplication (2)

Answer 2

(1) (a + b) + c = a + (b + c)

(2) (a · b) · c = a · (b · c)

Question 3

State the formula for the distributive property of the multiplication, first with respect to the addition (1) and then with respect to the subtraction (2)

Answer 3

(1) a · (b + c) = a · b + a · c
(2) a · (b - c) = a · b - a · c

Question 4

State the formula for the distributive property of the division, first with respect to the addition (1) and then with respect to the subtraction (2)

Answer 4

(1) (c + d) : a = c : a + d : a
(2) (c - d) : a = c : a - d : a

Question 5

State the formula for the invariance property of the subtraction. Use both an addition (1) and a subtraction (2) to show both cases

Answer 5

(1) a - b = (a+c) - (b+c)

(2) a - b = (a-c) - (b-c)

Question 6

State the formula for the invariance property of the division. Use both a multiplication (1) and a division (2) to show both cases

Answer 6

(1) a : b = (a·c) : (b·c)

(2) a : b = (a:c) : (b:c)

Question 7

The number a is such that a+b=b+a=b no matter the value of b. What is a?

Answer 7

a is the null element (elemento neutro) of the operation “+”.

Question 8

The number a is such that a·b=b·a=b no matter the value of b. What is a?

Answer 8

a is the null element (elemento neutro) of the operation “·”

Question 9

What does this example show?

(35 - 10) - 8 = 17

35 - (10 - 8) = 33

Answer 9

It shows that the associative property does not hold for the subtraction

Question 10

What does this example show?

120 : (30 : 2) = 8

(120 : 30) : 2 = 20

Answer 10

It shows that the associative property does not hold for the division

Question 11

Find the result of the following operations using the distributive property

1) 7 · 58 =
2) 39 : 3 =
3) 20 · 29 =

Answer 11

1) 7 · (50+8) = 350 + 56 = 406
2) 39 : 3 = (30 + 9) : 3 = 10 + 3 = 13
3) 20 · 29 = 20 · (30-1) = 600 - 20 = 580

Question 12

What properties have been used in the following calculations?

1) 300 : 15 = 60 : 3 = 20 : 1 = 20
2) 42 : 0.3 = 420 : 3 = 140
3) 3 · a · 2 = 6 · a
4) 31 - 28 = 30 - 27 = 10 - 7 = 3
5) (x + y) : x = 1 + y:x

Answer 12

1) Invariance of the division (dividing each term by 5, then by 3)
2) Invariance of the division (multiplying each term by 10)
3) commutativity of the multiplication
4) invariance of the subtraction (subtracting from each term 1 first, and then 20)
5) distributivity of the division with respect to the addition

Question 13

What is this?

“adding or subtracting the same quantity from each term of a subtraction does not change the result”

Answer 13

It describes the invariance property of the subtraction

Question 14

What is this?

“multiplying or dividing each term of a division by the same quantity does not change the result”

Answer 14

It is the invariance property of the division

Question 15

“When adding a set of three or more numbers together, one cangroup them in parenthesis in whichever way, and the result will not change.”

What is it?

Answer 15

It is the associative property of the addition

Question 16

What does PEDMAS stand for?

Answer 16

The order of operations:

Parentheses first,

Exponents,

Divisions and Multiplications,

Additions and Subtractions.

Question 17

Is this property valid?

a - (b - c) = a - b + c

Answer 17

YES, when a minus is in front of a parenthesis, it turns all the signs inside the parenthesis:

7 - (3 - 2) = 7 - 1 = 6

7 - 3 + 2 = 4 + 2 = 6

Question 18

The operation * has the following properties:

1) a*b=b*a
2) a*b*c = a*(b*c)

What does that mean?

Answer 18

That the operation * is both commutative and associative

Question 19

2³ = 8 , while 3² = 9

What does that mean?

Answer 19

The the operation “power” is not commutative, so the order matters!

Question 20

(2³)² = 8² = 64

while

2⁹=512

what does that mean?

Answer 20

That the operation “power” is not associative