# Problems from Operations with integers

Do the following calculations:

1. $$(+7)+(-3)=$$
2. $$(-5)+(-2)=$$
3. $$(+7)-(+2)=$$
4. $$(+9)-(-6)=$$
See development and solution

### Development:

• They have different signs
• We calculate the absolute values of each number: $$|+7|=7,$$ $$|-3|=3$$
• We subtract the absolute values: $$7-3=4$$
• We put the sign of the number with the greatest absolute value. In this case, $$7$$ is greater than $$-3$$, therefore we put the $$+$$ sign: $$+4$$
• And so, the result is: $$(+7)+(-3)=+4$$
• They have the same sign
• We calculate the absolute values of each number: $$|-5|=5$$, $$|-2|=2$$.
• We add up the absolute values: $$5+2=7$$
• We put the sign they had before: $$-7$$
• So the result is: $$(-5)+(-2)=-7$$
• The minuend is $$+7$$, and the subtrahend is $$+2$$.
• The opposite of $$+2$$ is $$-2$$.
• We add up the minuend ($$+7$$) and the opposite of the subtrahend ($$-2$$): $$(+7)+(-2)=+5$$
• The result of the subtraction is $$(+7)-(+2)=+5$$
• The minuend is $$+9$$, and the subtrahend is $$-6$$.
• The opposite of $$-6$$ is $$+6$$.
• We add up the minuend ($$+9$$) and the opposite of the subtrahend ($$+6$$): $$(+9)+(+6)=+15$$
• The result of the subtraction is $$(+9)-(-6)=+15$$

### Solution:

1. $$(+7)+(-3)=+4$$
2. $$(-5)+(-2)=-7$$
3. $$(+7)-(+2)=+5$$
4. $$(+9)-(-6)=+15$$
Hide solution and development

Do the following multiplications:

1. $$(+8)\cdot(+4)=$$
2. $$(+2)\cdot(-7)=$$
3. $$(-3)\cdot(-6)=$$
See development and solution

### Development:

1. We do the multiplication without signs: $$8\cdot4=32$$ As both numbers have the same sign, the result has a positive sign. That is: $$(+8)\cdot(+4)=+32$$

2. We do the multiplication without signs: $$2\cdot7=14$$ As both numbers have different signs, the result has a negative sign. That is: $$(+2)\cdot(-7)=-14$$

3. We do the multiplication without signs: $$3\cdot6=18$$ As both numbers have the same sign, the result has a positive sign. That is to say: $$(-3)\cdot(-6)=+18$$

### Solution:

1. $$(+8)\cdot(+4)=+32$$
2. $$(+2)\cdot(-7)=-14$$
3. $$(-3)\cdot(-6)=+18$$
Hide solution and development

Write the following expressions in a single power form:

1. $$(-2)^3 \cdot (-2)^5=$$
2. $$(+6)\cdot(+6)\cdot(+6)=$$
3. $$(+12)^4:(+12)^2=$$
4. $$\dfrac{1}{(+5)^{+3}}=$$
5. $$\big((-7)^4)\big)^4=$$
See development and solution

### Development:

1. It is a multiplication of two powers with the same base, therefore the exponents are added: $$(-2)^3 \cdot (-2)^5=(-2)^{3+5}=(-2)^8$$
2. $$6$$ is multiplied by $$3$$, so the power can be written as follows: $$(+6)\cdot(+6)\cdot(+6)=(+6)^3$$
3. It is a division of powers with the same base, therefore the exponents are subtracted: $$(+12)^4:(+12)^2=(+12)^{4-2}=(+12)^2$$
4. It is 1 divided by a power with a positive exponent, therefore it can be written as a power with negative exponent: $$\dfrac{1}{(+5)^{+3}}=(+5)^{-3}$$
5. It is a power of a power, and therefore we multiply the exponents: $$\big((-7)^4)\big)^4=(-7)^{16}$$

### Solution:

1. $$(-2)^3 \cdot (-2)^5=(-2)^8$$
2. $$(+6)\cdot(+6)\cdot(+6)=(+6)^3$$
3. $$(+12)^4:(+12)^2=(+12)^2$$
4. $$\dfrac{1}{(+5)^{+3}}=(+5)^{-3}$$
5. $$\big((-7)^4)\big)^4=(-7)^{16}$$
Hide solution and development

Do the following divisions:

1. $$(-21):(+3)=$$
2. $$(-64):(-8)=$$
3. $$(+50):(-10)=$$
See development and solution

### Development:

1. First we do the division without the signs: $$21:3=7$$ As both numbers have different signs, the result has a negative sign. Therefore: $$(-21):(+3)=-7$$

2. First we do the division without the signs: $$64:8=8$$ As both numbers have the same sign, the result has a positive sign. That is to say: $$(-64):(-8)=+8$$

3. We do the division without the signs: $$50:10=5$$ As the signs of the two numbers are different, the result is negative: $$(+50):(-10)=-5$$

### Solution:

1. $$(-21):(+3)=-7$$
2. $$(-64):(-8)=+8$$
3. $$(+50):(-10)=-5$$
Hide solution and development