# PEMDAS Rules and Operations on Real Numbers

Now that you have already learned the four fundamental operations on real numbers – addition, subtraction, multiplication, division – it is time to combine these operations into a single problem. In the Philippine Civil Service Examination, most of the problems on operations on real numbers have at least two or more operations involved. If you can recall, we call these operations MDAS in the elementary grades and later and PE making it PEMDAS. PEMDAS is the acronym for Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction. This is basically the order of operations when you calculate an arithmetic problem involving two or more operations.

*PEMDAS RULES*

Calculate in the following order.

- the expressions inside the
**P**arentheses. - the expression with
**E**xponents. - If no operation separates
**M**ultiplication and**D**ivision, perform from left hand side to right whichever comes first. - If no operation separates
**A**ddition and**S**ubtraction, perform from left hand side to right whichever comes first.

*EXAMPLES*

*Example 1: 4 + 3 x 5*

Perform multiplication first before addition since M comes before A in PEMDAS.

Multiply: 4 + 3 x 5 = 4 + 15

Add: 4 + 15 = 19.

*Example 2: (3 + 3) x 5*

Simplify the expression inside the parenthesis first before multiplying since P comes before M in PEMDAS.

Parenthesis: (3 + 3) x 5 = 6 x 5

Multiply: 6 x 5 = 30.

*Example 3: 8 + 4 ^{2} x 3*

Simplifying the expression with exponent first, the multiply, and then add.

Exponent: 8 + 4^{2} x 3* = *8 + 16 x 3

Multiply: 8 + 16 x 3 = 8 + 48

Add: 8 + 48 = 56

*Example 4: **3 x 4 + 6 x 2 – 5*

Perform multiplication simultaneously, add, and then subtract.

Multiply: 3 x 4 + 6 x 2 – 5 = 12 + 12 – 5

Add: 12 + 12 – 5 = 24 – 5

Subtract: 24 – 5 = 19

**Example 5: (4 + 5) x (8 – 2) ^{2} ÷ 2**

Perform the operations inside the parentheses simultaneously, simplify the operation with exponent, multiply, and then divide.

Parentheses: (4 + 5) x (8 – 2)^{2} *÷ *2 = 9 x 6^{2} ÷ 2

Exponent: 9 x 6^{2} ÷ 2 = 9 x 36 ÷ 2

Multiplication: 9 x 36 ÷ 2 = 324 ÷ 2

Divide: 324 ÷ 2 = 162

**Example 6: (5 + 8) ^{2} – 18 ÷ 6 x 2**

Parenthesis: (5 + 8)^{2} – 18 ÷ 6 x 2 = 13^{2 }– 18 ÷ 6 x 2

Exponent: 13^{2 }– 18 ÷ 6 x 2 = 169 – 18 ÷ 6 x 2

*Divide: 169 – 18 ÷ 6 x 2 = 169 – 3 x 2

*Multiply: 169 – 3 x 2 = 169 – 6

Subtract: 169 – 6 = 163

**COMMON MISTAKES in PEMDAS**

I have placed * in the division and multiplication operations in the last example above because this is one of the misconceptions of a lot of people. Many think that since M comes before D, multiplication must always be performed first before division.

That is NOT ALWAYS the case.

If multiplication and division operations are **NOT SEPARATED** by other operations or grouping symbols, you must simplify from the left hand side to the right hand side whichever operation comes first. For example, the expression 6 ÷ 3 x 2 has no +, – or () in between. So what is the answer? If your answer is 4, you are correct. Why?

Notice that if multiplication is performed first, then the expression becomes 6 ÷ 6 = 1. But the correct answer is 4 because 6 ÷ 3 = 2 and 2 x 2 = 4. If you don’t believe me, try to key in the expression 6 ÷ 3 x 2 in a calculator and then press the equal sign.

**Continuation:** PEMDAS Practice Test 1