Unit 1.2 Order of Operations (PEMDAS)

Examples

1: Simplify the expression 4(2·4+5)-8

For starters, let's look at what's inside the parentheses, because that is the first step. Inside we have 2·4+5, so we should first do multiplication, giving us 8+5, which we can simply add up to 13.

Now that we simplified what was inside the parentheses, we have the expression 4(13)-8 and in case you did not already know, a number in front of the parentheses indicates that the outer number should multiply what is inside of the parentheses, so we really have 4·(13)-8 so we can multiply 4 and 13 to get the product of 52.

Finally, we can subtract 8 from 52 and get our final answer of 44.

Answer: 44

2: Simplify the expression 9-4x3+6²

Firstly, we must apply the exponent to 6, so we now have the expression 9-4x3+36, due to 6² equaling 36.

Next, we can multiply 4 by 3 to get the product of 12, due to that being the next step in PEMDAS that applies to this equation, giving us the expression 9-12+36.

Finally, we can apply addition and subtraction from left to right, giving us the final answer of 33.

Anwer: 33

3: Simplify the expression 2÷3·9+(18+2÷4)

Now we're ramping up the difficulty a bit, and not grouping multiplication and division together in this one could cause you to get the wrong answer.

Looking inside the parentheses first off, we have to still utilize PEMDAS even inside of them, so we'll simplify 2÷4 to 0.5, and then we can simply add 18 to fully simplify our parentheses, giving us the equation 2÷3·9+18.5 (since parentheses are fully simplified, we can get rid of them.)

This is where not grouping multiplication/division would give you the wrong answer. If you simply multiplied first, then divided, you would have the expression 2÷27+18.5, which would result in the wrong answer. Instead, you must do multiplication and division in the same step, going from left to right, giving us ⅔, aka 0.66... · 9, which gives the product 6, which gives us the simplified expression 6+18.5, which we can simply add to get the final answer of 24.5.

Answer: 24.5

4:  Simplify the expression 9+(2²÷7-4)(6+4²-21)

We have two separate sets of exponents in this one, so let's simply those first. For our first set of parentheses, we can apply the exponent to 2 to get 4÷7-4, then we can apply division to get approximately 0.57-4, and then we can subtract 4 to now have the expression 9+(-3.43)(6+4²-21)

Simplifying the other set of parentheses, we can apply the exponent so the expression is 6+16-1, which we can then simply add/subtract from left to right to get 1.

Now, our expression is 9+(-3.43)(1). As said before, there is an "invisible" multiplication side between parentheses that can be inferred, so we can multiply the parentheses to now have 9+-3.43 due to anything multiplied by 1 being the exact same.

With an addition symbol then a subtraction symbol, you can cancel out the addition symbol due to adding a negative number being the same as subtracting, so we can finally just do the equation 9-3.43 to get a final answer of 5.57.

Answer: 5.57

5: Simplify the expression 4982÷(23-9+6²·√49)

As usual, let's first look inside the parentheses. We can simplify this rather easily with no awkward numbers. Firstly, we can square 6 to get 36, and then also get 7 as the square root of 49 due to 7 times 7 equaling 49. 

Next, we can then multiply 36 and 7 to get a product of 252, and finally apply addition and subtraction to get 266.

Finally, we can simply divide 4982 by 266 to get the quotient of around 18.72. 

Advice: Although sometimes you may get rather "ugly" answers in math, that never necessarily means that they are wrong. Your teachers may simply be challenging you, which is, of course, the best way to improve at anything. Additionally, even though an equation or expression may look overwhelming or tedious at first, the best strategy to follow is breaking it up into smaller pieces and then simplifying, similarly to how I am walking you through these examples.

Answer: 18.72

6: Simplify the expression 12[8÷(19-3·5)]+6²÷9(8²·42÷56)

Now you see why I gave you advice on the previous problem. The brackets are simply the same meaning as parentheses, but show to complete the inside set first. But as I said, we can simply break it up into smaller pieces. Firstly, we should look at the parentheses as usual. Since there is another set inside of it, we can first simplify that by multiplying 3 and 5 to get 15, then subtracting 15 from 19 to get the difference of 4. Then, we can take that set of parentheses away and divide 8 by 4 to get the quotient of 2. 

Moving on to the second set of parentheses. We can square 8 to get 64, then apply multiplication to get 2688÷56, then divide to get 48.

Now our expression looks a lot simpler, being 12(2)+6²÷9(48). We can then square the 6 to get 36, and all we have now is multiplication/division and addition. We can multiply 12 and 2 to get 24, divide 36 by 9 to get 4, and multiply 4 by 48 to get 192. Finally, we can add 24 and 192 to get a final answer of 216.

Answer: 216

1: Simplify the expression 9-2.5(3+12÷6)

2: Simplify the expression (2.5·8²)÷6+22

3: Simplify the expression 23-8÷2+(7·3³)

4: 34.6+(-9·4)+1

5: 3[9²÷3+9(4.4)]

6: 82+3(19-18÷6)-2³ 

7:−2(33·2+45+1)+98÷6

8: 23÷[7+8-(4·90÷28)]

9: -0.5(90+2²-37·8)+2

10: 16(3³÷54)·33-17

11: [2(3³·8²÷9)]·6+47÷1²

12: 4[52(8+48÷3)-51]÷(84÷7+15²)

1:  -3.5

2: 48.67

3: 208

4: -0.4

5: 199.8

6: 122

7: -207.67

8: 10.73

9: 103

10: 247

11: 2351

12: 20.2