Exercise 1

Write the following integers in increasing order, represent them graphically, and calculate the additive inverse and absolute values of the numbers:

8, -6, 5, 3, -2, 4, -4, 0, 7

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Exercise 2

Write the following integers graphically in increasing order and calculate the additive inverse and absolute values.

-4, 6, -2, 1, 5, 0, 9

Exercise 3

Remove the common factor in the following expressions:

1. 3 \times 2 + 3 \times (-5) =

2. (-2) \times 12 + (-2) \times (-6) =

3. 8 \times 5 + 8 = 8 \times (5 + 1)

4. (-3) \times (-2) + (-3) \times (-5) =

Exercise 4

Calculate:

1

(3 - 8) + [5 - (-2)] =

2

5 - [6 - 2 - (1 - 8) - 3 + 6] + 5 =

3

9 : [6 : (-2)] =

4

{ [{ (-2) }^{ 5 } - { (-3) }^{ 3 }] }^{ 2 } =

5

(5 + 3 \times 2 : 6 - 4) \times (4 : 2 - 3 + 6) :  { (7 - 8 : 2 - 2) }^{ 2 } =

6

[{ (17 - 15) }^{ 3 } + { (7 - 12) }^{ 2 }] : [(6 - 7) \times (12 - 23)] =

Exercise 5

Calculate:

1
(7 - 2 + 4) - (2 - 5) =

2
1 - (5 - 3 + 2) - [5 - (6 - 3 + 1) - 2] =

3
-12 \times 3 + 18 : (-12 : 6 + 8) =

Exercise 6

Solve, if it exists:

1

\sqrt { { (-9) }^{ 2 } } =

2

\sqrt { { (-1) }^{ 7 } } =

3

\sqrt { { (-3) }^{ 2 } \times (-3) } =

4

\sqrt { \frac { { (-2) }^{ 4 } }{ { (-2) }^{ 2 } } }

5

\sqrt { { (-3) }^{ 3 } } =

6

\sqrt { \frac { { (-8) }^{ 3 } }{ { (-2) }^{ 5 } } } =

Exercise 7

Calculate:

1

{ (-2) }^{ 2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

2

(-8) \times { (-2) }^{ 2 } \times { (-2) }^{ 0 } \times (-2) =

3

{ (-2) }^{ -2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

4

{ (2) }^{ -2 } \times { (2) }^{ -3 } \times { (2) }^{ 4 } =

5

{ (2) }^{ 2 } : { (2) }^{ 3 } =

6

{ (2) }^{ -2 } : { (2) }^{ 3 } =

7

{ (2) }^{ 2 } : { (2) }^{ -3 } =

8

{ (2) }^{ -2 } : { (2) }^{ -3 } =

9

{ [{ (-2) }^{ -2 }] }^{ 3 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

10

{ [{ (-2) }^{ 6 } : { (-2) }^{ 3 }] }^{ 3 } \times (-2) \times { (-2) }^{ -4 } =

Exercise 8

Calculate:

1

{ (-3) }^{ 1 } \times { (-3) }^{ 3 } \times { (-3) }^{ 4 } =

2

(-27) \times (-3) \times { (-3) }^{ 2 } \times { -3 }^{ 0 } =

3

{ (-3) }^{ 2 } \times { (-3) }^{ 3 } \times { (-3) }^{ -4 } =

4

{ (3) }^{ -2 } \times { (3) }^{ -4 } \times { (3) }^{ 4 } =

5

{ 5 }^{ 2 } : { 5 }^{ 3 } =

6

{ 5 }^{ -2 } : { 5 }^{ 3 } =

7

{ 5 }^{ 2 } : { 5 }^{ -3 } =

8

{ 5 }^{ -2 } : { 5 }^{ -3 } =

9

{ (-3) }^{ 1 } \times { [{ (-3) }^{ 3 }] }^{ 2 } \times { (-3) }^{ -4 } =

10

{ [{ (-3) }^{ 6 } : { (-3) }^{ 3 } ] }^{ 3 } \times { (-3) }^{ 0 } \times { (-3) }^{ -4 } =

 

Solution of exercise 1

Write the following integers in increasing order, represent them graphically, and calculate the additive inverse and absolute values of the numbers:

8, -6, 5, 3, -2, 4, -4, 0, 7

-6 < -5 < -4 < -2 < 0 < 3 < 4 < 7 < 8

op(-6) = -(-6) = 6 \qquad |-6| = 6

op(-6) = -(-6) = 6 \qquad |-6| = 6

op(-5) = -(-5) = 5 \qquad  |-5| = 5

op(-4) = -(-4) = 4 \qquad  |-4| = 4

op(-2) = -(-2) = 2  \qquad |-2| = 2

op(0) = 0  \qquad  |0| = 0

op(3) = -3   \qquad  |3| = 3

op(4) = -4 \qquad  |4| = 4

op(7) = -7 \qquad  |7| = 7

op(8) = -8 \qquad  |8| = 8

 

Solution of exercise 2

Write the following integers graphically in increasing order and calculate the additive inverse and absolute values.

-4, 6, -2, 1, 5, 0, 9

op(-4) = -(-4) = 4 \qquad  |-4| = 4

op(6) = -6 \qquad  |6| = 6

op(-2) = -(-2) = 2 \qquad  |-2| = 2

op(1) = - 1  \qquad  |1| = 1

op(-5) = -(-5) = 5 \qquad  |-5| = 5

op(0) = 0  \qquad  |0| = 0

op(9) = -9  \qquad |9| = 9

 

Solution of exercise 3

Remove the common factor in the following expressions:

1. 3 \times 2 + 3 \times (-5) =

= 3 \times [2 + (-5)] = 3 \times (2 - 5) = 3 \times (-3) = -9

 

2. (-2) \times 12 + (-2) \times (-6) =

= (-2) \times [12 + (-6)] = (-2) \times (12 - 6) = (-2) \times 6 = -12

 

3. 8 \times 5 + 8 = 8 \times (5 + 1)

= 8 \times 6 = 48

 

4. (-3) \times (-2) + (-3) \times (-5) =

= (-3) \times [(-2) + (-5)] = (-3) \times (-2 - 5) = (- 3) \times (-7) = 21

 

Solution of exercise 4

Calculate:

1

(3 - 8) + [5 - (-2)] =

(3 - 8) + [5 - (-2)] = -5 + (5 + 2) = -5 + 7= 2

 

2

5 - [6 - 2 - (1 - 8) - 3 + 6] + 5 =

5 - [6 - 2 - (1 - 8) - 3 + 6] + 5 =

= 5 - [6 - 2 - (-7) - 3 + 6] + 5 =

= 5 - [6 - 2 + 7 - 3 + 6] + 5 =

= 5 - 14 + 5 = -4

 

3

9 : [6 : (-2)] =

9 : [6 : (-2)] = 9 : (-3) = -3

 

4

{ [{ (-2) }^{ 5 } - { (-3) }^{ 3 }] }^{ 2 } =

{ [{ (-2) }^{ 5 } - { (-3) }^{ 3 }] }^{ 2 } =

= [-32 - (-27)] = { (-32 + 27) }^{ 2 } =

={ (-5) }^{ 2 } = 25

 

5

(5 + 3 \times 2 : 6 - 4) \times (4 : 2 - 3 + 6) :  { (7 - 8 : 2 - 2) }^{ 2 } =

(5 + 3 \times 2 : 6 - 4 ) \times (4 : 2 - 3 + 6) : { (7 - 8 : 2 - 2) }^{ 2 } =

= (5 + 6 : 6 - 4 ) \times (4 : 2 - 3 + 6) : { (7 - 8 : 2 - 2) }^{ 2 } =

= (5 + 1 - 4 ) \times (2 - 3 + 6) : { (7 - 8 : 2 - 2) }^{ 2 } =

= 2 \times 5 : { 1 }^{ 2 } =

= 2 \times 5 : 1 = 10 : 1 = 10

 

6

[{ (17 - 15) }^{ 3 } + { (7 - 12) }^{ 2 }] : [(6 - 7) \times (12 - 23)] =

[{ (17 - 15) }^{ 3 } + { (7 - 12) }^{ 2 }] : [(6 - 7) \times (12 - 23)] =

= [{ (2) }^{ 3 } + { (-5) }^{ 2 }] : [(-1) \times (-11)] =

= (8 + 25) : [(-1) \times (-11)] =

= (8 + 25) : 11 =

= 33: 11 = 3

 

Solution of exercise 5

Calculate:

1
(7 - 2 + 4) - (2 - 5) =

(7 - 2 + 4) - (2 - 5) = 9 - (-3) = 9 + 3 =12

 

2
1 - (5 - 3 + 2) - [5 - (6 - 3 + 1) - 2] =

1 - (5 - 3 + 2) - [5 - (6 - 3 + 1) - 2] =

= 1 - (4) - [5 - (4) - 2] =

= 1 - (4) - (5 - 4 - 2)=

= 1 - (4) - (-1) =

= 1 - 4 + 1 = −2

 

3
-12 \times 3 + 18 : (-12 : 6 + 8) =−12 · 3 + 18 : (−12 : 6 + 8) =

= -12 \times 3 + 18 : (-12 : 6 + 8) =

-12 \times 3 + 18 : (-2 + 8) =

= -12 \times 3 + 18 : 6 =

= -36 + 3 = -33

 

4

[( -12 + 36) : 6 + (8 - 5) : (-3)] - 6 =

= 2 \times [24 : 6 +3 : (-3)] − 6 =

= 2 \times [ 4 + (-1)] − 6 =

2 \times 3 - 6 = 6 - 6 = 0

 

5

[{ (-2) }^{ 5 } \times { (-3) }^{ 2 }] : { (-2) }^{ 2 } =

(-32 \times 9) : 4 = -288 : 4 = -72

 

6

6 + {4 - [(17 - (4 \times 4)] + 3} - 5 =

= 6 + {4 - [(17 - (4 \times 4)] + 3} - 5 =

6 + [4 - (17 - 16) + 3] - 5 =

= 6 + (4 - 1 + 3) - 5 =

6 + 6 - 5 = 7

 

Solution of exercise 6

Solve, if it exists:

1

\sqrt { { (-9) }^{ 2 } } =

\sqrt { { (-9) }^{ 2 } } = \sqrt { 81 } = \pm 9

 

2

\sqrt { { (-1) }^{ 7 } } =

\sqrt { { (-1) }^{ 7 } } = \sqrt { -1 } = no solution

3

\sqrt { { (-3) }^{ 2 } \times (-3) } =

\sqrt { { -3 }^{ 2 } \times (-3) } = \sqrt { { (-3) }^{ 3 } } = \sqrt { -27 } =no solution

 

4

\sqrt { \frac { { (-2) }^{ 4 } }{ { (-2) }^{ 2 } } }

\sqrt { \frac { { (-2) }^{ 4 } }{ { (-2) }^{ 2 } } } = \sqrt { { (-2) }^{ 2 } } = \sqrt { 4 } = \pm 2

 

5

\sqrt { { (-3) }^{ 3 } } =

\sqrt { { (-3) }^{ 3 } } = \sqrt { -27 } = no solution

 

6

\sqrt { \frac { { (-8) }^{ 3 } }{ { (-2) }^{ 5 } } } =

\sqrt { \frac { { (-8) }^{ 3 } }{ { (-2) }^{ 5 } } } = \sqrt { \frac { { { (-2) }^{ 3 } }^{ 3 } }{ { (-2) }^{ 5 } } } = \sqrt { \frac { { (-2) }^{ 9 } }{ { (-2) }^{ 5 } } } = \sqrt { { (-2) }^{ 4 } } = \sqrt { 16 } = \pm 4

 

Solution of exercise 7

Calculate:

1

{ (-2) }^{ 2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

{ (-2) }^{ 2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } = { (-2) }^{ 9 } = -512

 

2

(-8) \times { (-2) }^{ 2 } \times { (-2) }^{ 0 } \times (-2) =

(-8) \times { (-2) }^{ 2 } \times { (-2) }^{ 0 } \times (-2) =

= { (-2) }^{ 3 } \times { (-2) }^{ 2 } \times { (-2) }^{ 0 } \times (-2) = { (-2) }^{ 6 } = 64

 

3

{ (-2) }^{ -2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

{ (-2) }^{ -2 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } = { (-2) }^{ 5 } = -32

 

4

{ (2) }^{ -2 } \times { (2) }^{ -3 } \times { (2) }^{ 4 } =

{ 2 }^{ -2 } \times { 2 }^{ -3 } \times { 2 }^{ 4 } = { 2 }^{ -1 } = \frac { 1 }{ 2 }

 

5

{ (2) }^{ 2 } : { (2) }^{ 3 } =

{ 2 }^{ 2 } : { 2 }^{ 3 } = { 2 }^{ -1 } = \frac { 1 }{ 2 }

 

6

{ (2) }^{ -2 } : { (2) }^{ 3 } =

{ 2 }^{ -2 } : { 2 }^{ 3 } = { 2 }^{ -5 } = { (\frac { 1 }{ 2 }) }^{ 5 } = \frac { 1 }{ 32 }

 

7

{ (2) }^{ 2 } : { (2) }^{ -3 } =

{ 2 }^{ 2 } : { 2 }^{ -3 } = { 2 }^{ 5 } = 32

 

8

{ (2) }^{ -2 } : { (2) }^{ -3 } =

{ 2 }^{ -2 } : { 2 }^{ -3 } = 2

 

9

{ [{ (-2) }^{ -2 }] }^{ 3 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

{ [{ (-2) }^{ -2 }] }^{ 3 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } =

= { (-2) }^{ -6 } \times { (-2) }^{ 3 } \times { (-2) }^{ 4 } = -2

 

10

{ [{ (-2) }^{ 6 } : { (-2) }^{ 3 }] }^{ 3 } \times (-2) \times { (-2) }^{ -4 } =

{ [{ (-2) }^{ 6 } : { (-2) }^{ 3 }] }^{ 3 } \times (-2) \times { (-2) }^{ -4 } =

{ [{ (-2) }^{ 3 }] }^{ 3 } \times (-2) \times { (-2) }^{ -4 } =

={ (-2) }^{ 9 } \times (-2) \times { (-2) }^{ -4 } = { (-2) }^{ 6 } = 64

Solution of exercise 8

Calculate:

1

{ (-3) }^{ 1 } \times { (-3) }^{ 3 } \times { (-3) }^{ 4 } =

{ (-3) }^{ 1 } \times { (-3) }^{ 3 } \times { (-3) }^{ 4 } = { (-3) }^{ 8 } = 6561

 

2

(-27) \times (-3) \times { (-3) }^{ 2 } \times { -3 }^{ 0 } =

(-27) \times (-3) \times { (-3) }^{ 2 } \times { -3 }^{ 0 } = { (-3) }^{ 6 } = 729

 

3

{ (-3) }^{ 2 } \times { (-3) }^{ 3 } \times { (-3) }^{ -4 } =

{ (-3) }^{ 2 } \times { (-3) }^{ 3 } \times { (-3) }^{ -4 } = -3

 

4

{ (3) }^{ -2 } \times { (3) }^{ -4 } \times { (3) }^{ 4 } =

{ (3) }^{ -2 } \times { (3) }^{ -4 } \times { (3) }^{ 4 } = { 3 }^{ -2 } = { (\frac { 1 }{ 3 }) }^{ 2 } = \frac { 1 }{ 9 }

 

5

{ 5 }^{ 2 } : { 5 }^{ 3 } =

{ 5 }^{ 2 } : { 5 }^{ 3 } = { 5 }^{ -1 } = \frac { 1 }{ 5 }

 

6

{ 5 }^{ -2 } : { 5 }^{ 3 } =

{ 5 }^{ -2 } : { 5 }^{ 3 } = { 5 }^{ -5 } = { (\frac { 1 }{ 5 }) }^{ 5 } = \frac { 1 }{ 3125 }

 

7

{ 5 }^{ 2 } : { 5 }^{ -3 } =

{ 5 }^{ 2 } : { 5 }^{ -3 } = { 5 }^{ 5 } = 3125

 

8

{ 5 }^{ -2 } : { 5 }^{ -3 } =

{ 5 }^{ -2 } : { 5 }^{ -3 } = 5

 

9

{ (-3) }^{ 1 } \times { [{ (-3) }^{ 3 }] }^{ 2 } \times { (-3) }^{ -4 } =

{ (-3) }^{ 1 } \times { [{ (-3) }^{ 3 }] }^{ 2 } \times { (-3) }^{ -4 } = { (-3) }^{ 3 } = -27

 

10

{ [{ (-3) }^{ 6 } : { (-3) }^{ 3 } ] }^{ 3 } \times { (-3) }^{ 0 } \times { (-3) }^{ -4 } =

{ [{ (-3) }^{ 3 }] }^{ 3 } \times { (-3) }^{ 0 } \times { (-3) }^{ -4 } =

{ (-3) }^{ 9 } \times { (-3) }^{ 0 } \times { (-3) }^{ -4 } = { (-3) }^{ 5 } = -243

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.