Operations with integers

Problems

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. $$(+9)-(-6)=+15$$
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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. $$(+50):(-10)=$$
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. $$(+50):(-10)=-5$$
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