Exercises with Integers

1 -4, -1, 3, 2

Increasing order: − 4 < − 1 < 2 < 3

Decreasing order: 3 > 2 > -1 > -4

Graphical representation:

2 -4, -1, 0, -7

Increasing order: − 7 < − 4 < -1 < 0

Decreasing order: 0 > -1 > -4 > -7

Graphical representation:

3 2, 1, -1, 5, -3

Increasing order: − 3 < − 1 < 1 < 2 < 5

Decreasing order: 5 > 2 > 1 > -1 > -3

Graphical representation:

4 −4, 6, −2, 1, −5, 0, 9

Increasing order: − 5 < − 4 < − 2 < 0 < 1 < 6 < 9

Decreasing order: 9 > 6 > 1 > 0 > -2 > -4 > -5

Graphical representation:

5 8, −6, −5, 3, − 2, 4, −4, 0, 7

Increasing order: − 6 < − 5 < − 4 < − 2 < 0 < 3 < 4 < 7 < 8

Decreasing order: 8 > 7 > 4 > 3 > 0 > -2 > -4 > -5 > -6

Graphical representation:

1 -4, -1, 3, 2

Calculating the opposites:

-4 → -(-4)=4
-1 → -(-1)=1
3 → -(3)=-3
2 → -(2)=-2

Calculating the absolute values:

|-4|=4
|-1|=1
|3|=3
|2|=2

2 -4, -1, 0, -7

Calculating the opposites:

-4 → -(-4)=4
-1 → -(-1)=1
0 → -(0)=0
-7 → -(-7)=7

Calculating the absolute values:

|-4|=4
|-1|=1
|0|=0
|-7|=7

3 2, 1, -1, 5, -3

Calculating the opposites:

2 → -(2)=-2
1 → -(1)=-1
-1 → -(-1)=1
5 → -(5)=-5
-3 → -(-3)=3

Calculating the absolute values:

|2|=2
|1|=1
|-1|=1
|5|=5
|-3|=3

4 −4, 6, −2, 1, −5, 0, 9

Calculating the opposites:

-4 → -(-4)=4
6 → -(6)=-6
-2 → -(-2)=2
1 → -(1)=-1
-5 → -(-5)=5
0 → -(0)=0
9 → -(9)=-9

Calculating the absolute values:

|-4|=4
|6|=6
|-2|=2
|1|=1
|-5|=5
|0|=0
|9|=9

5 8, −6, −5, 3, − 2, 4, −4, 0, 7

Calculating the opposites:

8 → -(8)=-8
-6 → -(-6)=6
-5 → -(-5)=5
3 → -(3)=-3
-2 → -(-2)=2
4 → -(4)=-4
-4 → -(-4)=4
0 → -(0)=0
7 → -(7)=-7

Calculating the absolute values:

|8|=8
|-6|=6
|-5|=5
|3|=3
|-2|=2
|4|=4
|-4|=4
|0|=0
|7|=7

For this exercise we will use the distributive property: a(b+c)= a·b + a·c

1. 3 · 2 + 3 · (−5)

Direct solution: 3 · 2 + 3 · (−5) = 6 + (−15)= 6 − 15 = −9

Factorization: We can observe that the common factor is 3, we extract it using the distributive property:

3( 2 + (−5) )

Verification:

3(2+(−5)) = 3(2−5)=3(−3)= −9

2. (−2) · 12 + (−2) · (−6)

Direct solution: (−2) · 12 + (−2) · (−6) = −24 + 12 = −12

Factorization: We can observe that the common factor is −2, we extract it using the distributive property:

−2( 12 + (−6) )

Verification:

−2( 12 + (−6) ) = −2( 12−6)= −2(6)=−12

3. 8 · 5 + 8

Direct solution: 8 · 5 + 8 = 40 + 8 = 48

Factorization: We can observe that the common factor is 8, we extract it using the distributive property:

8( 5+1 )

Verification:

8( 5+1 ) =8(6)=48

4. 3 · 2 + 2

Direct solution: 3 · 2 + 2 = 6+ 2 = 8

Factorization: We can observe that the common factor is 2, we extract it using the distributive property:

2( 3+1 )

Verification:

2( 3+1 ) =2(4)=8

5. (−3) · (−2) + (−3) · (−5)

Direct solution: (−3) · (−2) + (−3) · (−5) = 6 + 15 = 21

Factorization: We can observe that the common factor is −3, we extract it using the distributive property:

−3( (−2)+(−5) )

Verification:

−3( (−2)+(−5) ) = −3(−2−5)= −3(−7)=21

1. 2 − (−3) −(−1) =

We write the opposite of (−3) and (−1)

= 2 + 3+1 =

We perform the addition

= 2+3+1 = 6

2. −4 − 3 −(−5) =

We write the opposite of (−5)

= −4 − 3+5 =

We perform the addition and subtraction

= −4−3+5 = −2

3. (3 − 8) + [5 − (−2)] =

We write the opposite of (−2)

= −5 + (5 + 2) =

We operate in the parentheses

= −5 + 7 = 2

4. 5 − [6 − 2 − (1 − 8) − 3 + 6] + 5 =

We operate in parentheses

= 5 − [6 − 2 − (−7) − 3 + 6] + 5 =

We calculate the opposite of (−7)

= 5 − (6 − 2 + 7 − 3 + 6) + 5 =

We operate in the parentheses and take the opposite of the result

= 5 − 14 + 5 = −4

5. [12 : 2] : 3 =

We perform the division in the brackets

= [12 : 2]:3 = 6:3

We perform the remaining division

= 6:3=2

6. 9 : [6 : (−2)] =

We perform the division in the brackets

= 9 : [6 : (−2)]=9 : (−3)

We perform the remaining division

= 9 : (−3)=−3

7. [24 : (−3)]: [16 : (− 4)] =

We perform the division in the brackets

= [24 : (−3)]: [16 : (− 4)]=−8 : (−4)

We perform the remaining division

= −8 : (−4)=2

8. [(−2)⁵ − (−3)³]² =

We perform the powers in the parentheses. For (−2)⁵ we calculate as follows: (−2)(−2) = 4. 4(−2) = −8. −8 (−2) = 16. 16 (−2) = −32. For (−3)³, following the same way of calculating, we have (−3)(−3) = 9. 9 (−3) = −27

= [−32 − (−27)]² =

We remove parentheses

= (−32 + 27)² =

We perform the operation and square

= (−5)² = 25

9. (5 + 3 · 2 : 6 − 4 ) · (4 : 2 − 3 + 6) : (7 − 8 : 2 − 2)² =

We perform 3 · 2 (we will do the multiplication first) and the divisions

= (5 + 6 : 6 − 4 ) · (2 − 3 + 6) : (7 − 4 − 2)² =

We perform the division

= (5 + 1 − 4 ) · (2 − 3 + 6) : (7 − 4 − 2)² =

We operate in each parenthesis

= 2 · 5 : 1² =

We square

= 2 · 5 : 1 =

First we have to multiply and then we will divide

= 10 : 1 =10

10. [(17 − 15)³ + (7 − 12)²] : [(6 − 7) · (12 − 23)]

We operate in the parentheses

= [(2)³ + (−5)²] : [(−1) · (−11)]

We perform the powers

= (8 + 25) : [(−1) · (−11)] =

We operate in the parentheses and in the brackets and divide the results

= 33 : 11 = 3

1. 10:5+2 =

We perform the division

= 10:5+2 = 2+2

We perform the addition

= 2+2 = 4

2. 10+6:3 =

We perform the division

= 10+6:3 = 10+2

We perform the addition

= 10+2 = 12

3. 6 · 3+6:3 =

We perform the multiplication and division

= 6 · 3+6:3 = 18+2

We perform the addition

= 18+2 = 20

4. 18:6− 4· 3 =

We perform the multiplication and division

= 18:6− 4· 3 = 3−12

We perform the subtraction

= 3−12 = −9

5. (7 − 2 + 4) − (2 − 5) =

We operate in the parentheses

= 9 − (−3) =

We take the opposite of (−3)

= 9 + 3 = 12

6. 1 − (5 − 3 + 2) − [5 − (6 − 3 + 1) − 2] =

We operate in the parentheses

= 1 − (4) − [5 − (4) − 2] =

We take the opposites of the two 4s

= 1 − 4 − (5 − 4 − 2) =

We operate in the parentheses

= 1 − 4 − 5 + 4 + 2 =

We add 4 with its inverse

= 1 − 5 + 2 = −2

7. −12 · 3 + 18 : (−12 : 6 + 8) =

We perform the division in the parentheses

= − 12 · 3 + 18 : (−2 + 8) =

We operate in the parentheses

= −12 · 3 + 18 : 6 =

We perform the multiplication and division

= −36 + 3 = −33

8. 2 · [( −12 + 36) : 6 + (8 − 5) : (−3)] − 6 =

We perform the operations in the first two parentheses

= 2 · [24 : 6 + 3 : (−3)] − 6 =

We perform the divisions

= 2 · [ 4 + (−1)] − 6 =

We operate in the brackets

= 2 · 3 − 6 =

We perform the product

= 6 − 6 = 0

9. [(−2)⁵ · (−3)²] : (−2)² =

We perform the powers

= (−32 · 9) : 4 =

We operate in the parentheses

= −288 : 4 = −72

10. 6 + {4 − [(17 − (4 · 4)] + 3} − 5 =

We perform the product

= 6 + [4 − (17 − 16) + 3] − 5 =

We operate in the parentheses and take its opposite

= 6 + (4 − 1 + 3) − 5 =

We operate in the parentheses

= 6 + 6 − 5 = 7

1 (−2)² = 4

The square root of 4 is ± 2

2 (−4)² = 16

The square root of 16 is ± 4

3 (−9)² = 81

The square root of 81 is ± 9

4 (−5)³ = −125

The square root of −125 does not exist in the reals, therefore this root has no solution in the set of real numbers.

In general, the square root of a negative number does not exist because there is no number that when squared has a negative sign.

5 (−3)⁵ = −243

The square root of −243 does not exist in the reals, therefore this root has no solution in the set of real numbers.

In general, the square root of a negative number does not exist because there is no number that when squared has a negative sign.

6 (−1)⁷ =  −1

The square root of −1 does not exist in the reals, therefore this root has no solution in the set of real numbers.

In general, the square root of a negative number does not exist because there is no number that when squared has a negative sign.

7 (−3)² · (−3) =

We perform the product of the powers and cube

  = (−3)³ = −27

The square root of −27 has no solution.

8

We perform the division of powers, square, and extract the root.

The result is 2, and the square root of 2 does exist in the real numbers.

9 (−2)³ = −8

The square root of −8 has no solution in the reals.

10

We put 8 in the form of a power, perform the power of a power in the numerator, divide the powers, raise to the fourth power, and extract the root.

The result is 4, and the square root of 4 is ±2.

To solve these exercises we will use the laws of exponents:

1 (−2)² · (−2)³ · (−2)⁴ = (−2)⁹ = −512

The result will have a negative sign because the base is negative and the exponent is odd.

2 (−8) · (−2)² · (−2)⁰ (−2) =

First we decompose 8 into factors

(−2)³ · (−2)² · (−2)⁰ · (−2) = (−2)⁶ = 64

The result will have a positive sign because the base is negative and the exponent is even.

3 (−2)⁻² · (−2)³ · (−2)⁴ = (−2)⁵ = −32

4 2⁻² · 2⁻³ · 2⁴ = 2⁻¹ = 1/2

Since the exponent is negative, we have to take the inverse of the base, that is, apply the corresponding sign law.

5 2² : 2³ = 2⁻¹ = 1/2

Since the exponent is negative, we have to take the inverse of the base, that is, apply the corresponding sign law.

6 2⁻² : 2³ = 2⁻⁵ = 1/2⁵ = 1/32

7 2² : 2⁻³ = 2⁵ = 32

8 2⁻² : 2⁻³ = 2

9 [(−2)⁻²]³ · (−2)³ · (−2)⁴ = (−2)⁻⁶ · (−2)⁷ = −2

10 [(−2)⁶ : (−2)³] ³ · (−2)· (−2)⁻⁴ = [(−2)³]³ · (−2)⁻³ = (−2)⁹ · (−2)⁻³= (−2)⁶ = 64

1 (−3)¹ · (−3)³ · (−3)⁴ = (−3)⁸ = 6561

2 (−27) · (−3) · (−3)² · (−3)⁰= (−3)³ · (−3) · (−3)² · (−3)⁰ = (−3)⁶ = 729

3 (−3)² · (−3)³ · (−3)⁻⁴ = −3

4 3⁻² · 3⁻⁴ · 3⁴ = 3⁻² = 1/3² = 1/9

5 5² : 5³ = 5⁻¹ = 1/5

6 5⁻² : 5³ = 5⁻⁵ = 1/5⁵ = 1/3125

7 5² : 5⁻³ = 5⁵ = 3125

8 5⁻² : 5⁻³ = 5

9 (−3)¹ · [(−3)³]² · (−3)⁻⁴ = (−3)¹ · (−3)⁶· (−3)⁻⁴ = (−3)³

First we calculate the power of a power and then we multiply

10 [(−3)⁶ : (−3)³]³ · (−3)⁰ · (−3)⁻⁴ = [(−3)³]³ · (−3)⁰· (−3)⁻⁴ = (−3)⁹ · (−3)⁰ · (−3)⁻⁴ = (−3)⁵ =−243

First we do the division indicated in the bracket, then we perform the power of a power and finally we multiply the powers