## Math 6 Chapter 2 Lesson 12: Properties of multiplication

## 1. Summary of theory Tóm

### 1.1. Commutative property

a . b = b . a

**Example 1: **Calculate

a. 2 . (-3)

b. (-7) . (-4)

__Solution guide__

a. 2 . (-3) = (-3) . 2 = (-6)

b. (-7) . (-4) = (-4) . (-7) = 28

### 1.2. Combined properties

(a.b) . c = a . (b.c)

**Example 2:** Calculate [9 . (-5)] . 2

__Solution guide__

[9 . (-5)] . 2 = 9 . [(-5) . 2] = -90

**Attention:**

– Thanks to the associative property, we can talk about the product of three, four, five, ..integers

For example: a . b . c = a . (b . c) = (a . b) . c

– When performing multiplication of many integers, we can rely on commutative and associative properties to change the position of factors, put brackets to group soos factors arbitrarily.

– We also call the product of n integers a to be the nth power of integer a (reading and notation as for natural numbers).

**For example:** (-2) . (-2) . (-2) =(-2)^{3}

**Comment: **In a product of non-zero integers:

a. If there is an even number of negative integer factors, the product has a “+” sign.

b. If there is an odd number of negative integer factors, the product has a “-” sign.

### 1.3. Multiply by 1

a . 1 = 1 . a = a

### 1.4. Distributive property of multiplication over addition

a(b+c) = ab + ac

**Attention:**

The above property is also true for subtraction: a(b – c) = ab – ac

**Example 3: **Replace a factor with a sum to calculate

a. -53 . 21

b. 45 . (-twelfth)

__Solution guide__

a. -53 . 21

= -53 . (20 + 1)

= -53 . 20 – 53 . first

= – 1060 – 53 = – 1113

b. 45 . (-twelfth)

= 45 . (-10) + 45. (-2)

= -450 – 90

= – 540

## 2. Illustrated exercise

**Question 1: **Calculate

a. (26 – 6) . (-4) + 31 . (-7 – 13)

b. (-18) . (55 – 24) – 28 . (44 – 68)

__Solution guide__

a. (26 – 6) . (-4) + 31 . (-7 – 13)

= 20 . (-4) + 31 . (-20)

= -20 .(4 + 31)

= -20.35

= -700

b. (-18) . (55 – 24) – 28 . (44 – 68)

= -18 . (31) – 28 . (-24)

= -558 + 672

= 114

**Verse 2: **Quick calculation

a. (-4) . (+3) . (-125) . (+25) . (-8)

b. (-sixty seven) . (1 – 301) – 301 . sixty seven

__Solution guide__

a. (-4) . (+3) . (-125) . (+25) . (-8)

= [(-4) . (+25)] . [(-125) . (-8)] . (+3)

= (-100) . (+1000) . (+3)

= -300 000

b. (-sixty seven) . (1 – 301) – 301 . sixty seven

= (-67) . 1 + 67 . 301 – 67 . 301

=- 67

**Question 3:** Write the following products as powers of an integer:

a. (-8) . (-3)3 . (+125)

b. 27 . (-2)3 . (-7) . (+49)

__Solution guide__

a. (-8) . (-3)3 . (+125)

= [(-2) . (-2) . (-2)] .[(-3) . (-3) . (-3)] . (5 . 5 . 5)

= [(-2) . (-3) . 5] . [(-2) . (-3) . 5] . [(-2) . (-3) . 5]

= 30 . 30 . 30

= 30^{3}

b. 27 . (-2)3 . (-7) . (+49)

= [3. 3 . 3] . [(-2) . (-2) . (2)] . (-7) . [(-7) . (-7)

=[3 .(-2) . (-7)] . [3 . (-2) . (-7)] . [3. (-2) . (-7)]

= 42 . 42 . 42

= 42^{3}

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Do the following calculations

a) (-4.2.6.25.(-7.5)

b) 16(38 – 2) – 38(16 – 1)

**Verse 2: **Prove that for a, b, c ∈ Z then:

a(b + c) – b(a + c) = b(a – c) – a(b – c)

### 3.2. Multiple choice exercises Bài

**Question 1:** Fast calculation (-5).125.(-8).20.(-2) we get the result:

A. -200000

B. -2000000

C. 200000

D. -100000

**Verse 2:** Expression value M = (-192873).(-2345).(-4)^{5}.0 is:

A. -192873

B. 1

C. 0

D. (-192873).(-2345).(-4)^{5}

**Question 3: **The value of the expression 27.(-13) + 27.(-27) + (-14).(-27) is:

A. – 702

B. 702

C. – 720

D. 720

**Question 4:** Fill in the next two numbers in the following sequence: -2; 4; -8; 16; …

A. 32 and 64

B. – 32 and 64

C. 32 and – 64

D. – 32 and – 64

**Question 5: **Fill in the correct number…: 17.(… + 7) = 17.(-5) + 17.7

A. – 2

B. – 3

C. – 4

D. – 5

**Question 6:** Which of the following statements is true?

A. When multiplying a negative number by two positive numbers, the result is a positive number

B. When you multiply two negative numbers by a positive number, the result is a negative number

C. When we multiply two negative numbers by two positive numbers, the result is a positive number

D. When multiplying a negative number by three positive numbers, the result is a positive number

## 4. Conclusion

Through this lesson, you should know the following:

- Master the properties of multiplication: commutative property, associative property, multiplying by 1, distributive property of multiplication with addition.
- Apply to do calculation exercises.

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