Elimination of Parentheses in Real Numbers

The logic in this case is that the subtraction sign allows us to obtain the number opposite to the one given to us.
Consequently:

• "Minus minus six" is equal to "plus six", that is, "six".
• Similarly, "minus plus six" is equal to "minus six".

However, the plus sign does not indicate a modification in the number.
Therefore,

• "plus minus six" is equal to "minus six"
• and "plus plus six" is equal to "plus six", that is, "six".

Examples:

• $(+50)+(-20) = (+30)$
• $(-8)-(+2) = (-10)$
• $(-3)-(-4) = (+1)$

Let's look again at the three previously solved exercises, now we will write them without parentheses.

• $(+50)+(-20) = (+30)$
  $50-20 = 30$
   
• $(-8)-(+2) = (-10)$
  $-8-2 = -10$
   
• $(-3)-(-4)=(+1)$
  $-3+4 = 1$

As we surely remember from the class on "real numbers", when a number has no sign, we understand it to be positive.
Therefore,

• in the first exercise, we can write "50" and "30" instead of "+50" and "+30".
• However, we cannot remove the plus sign in the third exercise: in "+4".

Remember: We can only omit the plus sign if the number is the first in the sequence.

When solving exercises with real numbers, in the first phase we will remove the parentheses according to mathematical rules.

Example:

$(+58)-(-34)+(+9)-(+5)+(-2) =$
$58+34+9-5-2 = 94$

Complete:

• $–(-10) =$ __
• $+(+8) =$ __
• __$(-9) = 9$
• __$(-9) = -9$
• __$+(-5) = -5$
• $($__$3) = -3$
• -$($__$) = 20$

Solve the following exercises, first of all, remove the parentheses:

• $(-5)+(+35)-(-22) =$
• $(-9)-(+2)+(+10) =$
• $(+56)+(-43)-(-4)-(+5) =$
• $(-12.8)-(-3.7)-(+5) =$
• $(-90)+(+4.7)-(-2.2) =$

Assignment

Mark the correct answer

$[(3-2+4)^2-2^2]:\frac{(\sqrt{9}\cdot7)}{3}=$

Solution

We solve the expressions inside the parentheses according to the order of arithmetic operations

$[(1+4)^2-2^2]:\frac{(3\cdot7)}{3}=$

We continue solving the expressions inside the parentheses accordingly.

$[5^2-2^2]:\frac{21}{3}=$

$[25-4]:\frac{21}{3}=$

$21:7=3$

Answer

$3$

Assignment

$(7+2+3)(7+6)(12-3-4)=\text{?}$

Solution

First, we solve the expressions within the parentheses according to the laws of addition and subtraction

$\left(9+3\right)\left(7+6\right)\left(9-4\right)=?$

$12\times13\times5=\text{?}$

We arrange the multiplication exercise we obtained to make it easier to solve.

$12\times5\times13=\text{?}$

We solve the exercise from left to right

$12\times5=60$

$60\times13=780$

Answer

$780$

Assignment

$\left(9+7+3\right)\left(4+5+3\right)\left(7-3-4\right)$

Solution

First, we solve the expressions within the parentheses according to the laws of addition and subtraction

$\left(9+10\right)\left(9+3\right)\left(4-4\right)=$

$19\times12\times0=$

Note that we obtained a multiplication exercise with the number $0$ and we solve it first to simplify the calculation.

$12\times0=0$

$19\times0=0$

Answer

$0$

Assignment

$(8-3-1)\times4\times3=$

Solution

First, we solve the operations inside the parentheses according to the rules of addition and subtraction

$(5-1)\times4\times3=$

$4\times4\times3=$

We solve the operation from left to right

$4\times4=16$

$16\times3=48$

Answer

$48$

Assignment

$(7+2)\times(3+8)=$

Solution

We multiply the first element inside the parentheses by the elements of the second parentheses

Then we multiply the second element inside the primary parentheses by the elements of the second parentheses

$7\times3+7\times8+2\times3+2\times8=$

We solve all the multiplication exercises from left to right

$21+56+6+16=$

Now we add from left to right

$21+56=77$

$77+6=83$

$83+16=99$

Answer

$99$