1. (9 + 3) - (2 + 4) = 6
2. [(16 - 4)/2] + 3 = 9.
3. 4^2 - [5 x (2 + 1)] = 1
4. 3 x (4 + 5) - 6 + 7 = 28
5. -6^2 is -36 and (-6)^2 is 36.
Step-by-step explanation:
In earlier math, you are usually given two numbers and an operator (addition, subtraction, multiplication, and division) and are asked to solve for the answer. Whether it be (1 + 2) or (91 x 82), you could solve it because that's the extent that it could go. But what if we had multiple numbers and operators, like (6 + 2 - 4) or ((6 / 2) * 3)?
All of the examples given are known as expressions. They are "numbers, symbols and operators grouped together that show the value of something." And as the bigger the expressions get, the bigger the questions are asked.
For example, take expressions (1 + 2 + 3) and (4 + 8 / 2). If you performed these expressions from left to right like you always did, you'd say that both equal 6! Sadly, they don't. So now you ask this question: How do I solve this expression? From left to right? Right to left? Where do I start? Mathematicians sturggled with this questions as well, so they all agreed that expressions be performed under an order of operations.
The order of operations is a set of rules used to solve a mathematical expression in an orderly manner. GIven a mathematical expression, you would have to do the following steps:
1. Grouping symbols, specifically the contents with them, are simplified.
2. Exponents are simplified.
3. Multiply or divide from left to right.
4. Add or subtract from left to right.
These sets of steps can also be known as GEMA.
So now that we know how to solve expressions, we can say that (1 + 2 + 3) is NOT equal to (4 + 8 / 2) because according to GEMA, we divide, then add.
1 + 2 + 3 =? (4 + 8 / 2)
  3 + 3  =? (4 + 4)
     6  
8
So now we can solve problems 1-5 with GEMA in mind.
For problem 1, we don't have to add any grouping symbols because the expression already equals 6, but if we had to, we can say that
(9 + 3) - (2 + 4) = 6.
       12 - 6 = 6
           6 = 6
For problem 2, we can abuse the rule that division precedes addition to change this expression. We can say that
[(16 - 4)/2] + 3 = 9.
    [12/2] + 3 = 9
       6 + 3 = 9
          9 = 9
Note that (,),[, and ] are grouping symbols. [ and ] precede ( and ) in the order of operations.
For problem 3, we can say that
4^2 - [5 x (2 + 1)] = 1.
     16 - [5 x 3] = 1
        16 - 15 = 1
For problem 4, we can say that
3 x (4 + 5) - 6 + 7 = 28.
    3 x 9 - 6 + 7 = 28
      27 - 6 + 7 = 28
         21 + 7 = 28
            28 = 28
For problem 5, -6^2 and (-6)^2 are totally different.
-6^2 means -1 * 6^2.
-6^2 = -1 * 6^2
    = -1 * 36
    = -36
(-6)^2 means (-6)(-6).
(-6)^2 = (-6)(-6)
     = 36
The rest is up to you. Good luck.
