**Order of Operations (or PEMDAS Rule)**

The key concept behind the order of operations is to perform arithmetic operators in the “right” order or sequence. Let’s take a look at how Rob and Patty tried to simplify a given numerical expression by applying the order of operations.

**What is Rob’s mistake?**

- He carelessly simplified the numerical expressions by applying arithmetic operations from left to right.

**Patty got the correct answer because she properly applied the rules on order of operations.**

- She performed multiplication and division first before addition and subtraction.

Let’s take a look at some examples…

**Example 1:** Simplify **5 ÷ 5 + 3 − 6 x 2** using the rules for order of operations

- Divide

= **5 ÷ 5** + 3 − 6 x 2

- Multiply

=1 + 3 − **6 x 2**

- Add

=**1 + 3** − 12

- Subtract

= 4 − 12

- Final answer

= −8

**Example 2:** Simplify **3 x 7 − 11 + 15 ÷ 3** using the rules for order of operations

- Multiply

= **3 x 7** − 11 + 15 ÷3

- Divide

= 21− 11 + **15 ÷3**

- Subtract

= **21− 11** + 5

- Add

= 10 + 5

- Final answer

= 15

The next examples will now involve parentheses. Remember that you have to simplify everything inside the parenthesis first before going forward. The rules for order of operations apply the same way inside the parenthesis.

**Example 3:** Simplify **25 − (7 − 12 ÷ 6 ) x 4** using the rules for order of operations

- Simplify everything inside the parenthesis first

= 25 − **(7 − 12 ÷ 6)** x 4

- Divide

= 25 − (7 − **12 ÷ 6**) x 4

- Subtract

= 25 − (**7 −2**) x 4

- Multiply

= 25 − **(5) x 4**

- Subtract

= 25 −20

- Final answer

= 5

**Example 4:** Simplify **5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5** using PEMDAS rule.

- Simplify the expressions inside the parentheses first

= 5**(4 + 3 x 2)** − **(9 −28 ÷ 7)** ÷ 5

- Multiply and Divide

= 5(4 + **3 x 2**) − (9 −**28 ÷ 7**) ÷ 5

- Add and Subtract

= 5(**4 + 6**) − (**9 − 4**) ÷ 5

- Multiply

= **5(10)** − (5) ÷ 5

- Divide

= 50 − **(5) ÷ 5**

- Subtract

= 50 −1

- Final answer

= 49

The final examples will involve exponents. Be careful in each step because they are so many things going on. As long as you remain focus in following the rules governing the order of operations, it shouldn’t be that difficult! Here we go…

**Example 5:** Simplify **2 ^{4}−5(10−4^{2} ÷ 2) + (30−3^{3})** using PEMDAS rule.

- Simplify the expressions inside the parentheses and the exponential numbers first.

= 2^{4}− 5 **(10 − 4 ^{ 2} ÷ 2 )** +

**(30− 3**

^{3 })- Exponents

= **2 ^{4}**− 5 (10 −

**4**÷ 2 ) + (30−

^{2}**3**)

^{3}- Divide

= 16 − 5 (10 −**16÷ 2** ) + (30−27)

- Subtract

= 16 − 5 (**10 − 8**) + (**30−27**)

- Multiply

= 16 − **5 (2)** + (3)

- Subtract

= **16 − 10** + (3)

- Add

= 6 + (3)

- Final answer

= 9

**Example 6:** Simplify **(32 − 3 ^{ 3 } ÷ 9 x 10)^{5 }− 4^{2} ÷ 8 + 3^{2} ** using PEMDAS rule.

- Simplify the expressions inside the parenthesis and the numbers with exponents first

=(32 − 3 ^{3 } ÷ 9 x 10 )^{5 }− 4^{2} ÷ 8 + 3^{2}

- Exponents

=(32 − **3 ^{3}** ÷ 9 x 10 )

^{5 }−

**4**÷ 8 +

^{2}**3**

^{2}- Divide

=(32 − **27 ÷ 9** x 10 )^{5 }−16÷ 8 + 9

- Multiply

=(32 − **3 x 10** )^{5 }−16÷ 8 + 9

- Subtract

=(**32 −30**)^{5 }−16÷ 8 + 9

- Exponent

=**(2) ^{5}**

^{ }−16÷ 8 + 9

- Divide

=32 −**16÷ 8** + 9

- Subtract

=**32 −2**+ 9

- Add

= 30+ 9

- Final answer

= 39