# Operations with integers

## Problems

Do the following calculations:

1. $(+7)+(-3)=$
2. $(-5)+(-2)=$
3. $(+7)-(+2)=$
4. $(+9)-(-6)=$
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### 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. $(+9)-(-6)=+15$
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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. $(-3)\cdot(-6)=+18$
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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. $\big((-7)^4)\big)^4=(-7)^{16}$
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Do the following divisions:

1. $(-21):(+3)=$
2. $(-64):(-8)=$
3. $(+50):(-10)=$
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### 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. $(+50):(-10)=-5$
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