Order of Operations: Algebra, Fraction, Exponents & Example

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Order of operations rule

When looking at an equation with multiple operations you can follow the four steps to help you calculate the equation in the correct order:

First you can start by calculating anything that is inside grouping symbols, like brackets or parentheses.
Now you look at any powers that are in the sum, these are also known as exponents.
Next you move on to any multiplication or division, working from left to right.
Finally, you work through any addition or subtraction, again working from left to right.

There are different types of brackets that may be used;

Round brackets ( )
Curly brackets { }
Box brackets [ ]

Solve $2 (3 + 10) + 4^{2}$

To solve this, you can break it down into the 4 steps;

Calculate grouping symbols : $(3 + 10) = 13$
Calculate any exponents : $4^{2} = 16$
Multiplication/division : $2 \times 13 = 26$
Addition/subtraction : $26 + 16 = 42$

Therefore, the answer is 42.

How to remember the rule?

In order to help you remember which order to solve a sum there is an acronym, PEMDAS;

P : Parentheses (grouping symbols)

E : Exponents (powers and roots)

MD : Multiplication and division

AS : Addition and subtraction

Order of operations examples

Simplify $5^{3} - (4 \times 2)$

Let's work through each step to solve the expression;

P : $4 \times 2 = 8$
E : $5^{3} = 125$
M/D : This step is not needed in this sum.
A/S : $125 - 8 = 117$

Therefore, $5^{3} - (4 \times 2) = 117$

Simplify $3 \times 4 + 7^{2}$

Let's work through each step to solve the expression;

P : This step is not needed in this problem.
E : $7^{2} = 49$
M/D : $3 \times 4 = 12$
A/S : $12 + 49 = 61$

Therefore, $3 \times 4 + 7^{2} = 61$

Order of operations with algebra

Sometimes when you are evaluating an algebraic expression for a given value of the variable, you will need to apply the Order of Operations to help you get the correct answer.

Evaluate $4 - x^{2} \times 3$ when $x = 6$

First you can substitute 6 into the expression, then you can follow PEMDAS;

4 - x^{2} \times 3 4 - 6^{2} \times 3

P : This step is not needed in this problem.
E : $6^{2} = 36$
M/D : $36 \times 3 = 108$
A/S : $4 - 108 = - 104$

Therefore when $x = 6$ , $4 - x^{2} \times 3 = - 104$

Evaluate $x^{3} - y \times (x + x)$ when $x = 6$ and $y = 3$

To begin with, you can substitute in your variables to see the expression, then you can continue to solve it using PEMDAS;

$x^{3} - y \times (x + x)$

$6^{3} - 3 \times (6 + 6)$

P : $(6 + 6) = 12$
E : $6^{3} = 216$
M/D : $3 \times 12 = 36$
A/S : $216 - 36 = 180$

Therefore, when $x = 6$ and $y = 3$ , $x^{3} - y \times (x + x) = 180$

Evaluate $x + 4 \times \frac{1}{2}$ when $x = 12$

To begin with, you can substitute in your variable to see the expression, then you can continue to solve it using PEMDAS;

$12 + 4 \times \frac{1}{2}$

P : This step is not needed in this problem.
E : This step is not needed in this problem.
M/D : $4 \times \frac{1}{2} = 2$
A/S : $12 + 2 = 14$

Therefore, when $x = 12$ , $x + 4 \times \frac{1}{2} = 14$

Evaluate $5^{2} + 12 \times {3 + x}$ when $x = 2$

To begin with, you can substitute in your variable to see the expression, then you can continue to solve it using PEMDAS;

$5^{2} + 12 \times {3 + 2}$

P : $3 + 2 = 5$
E : $5^{2} = 25$
M/D : $12 \times 5 = 60$
A/S : $25 + 60 = 85$

Therefore, when

x = 2

5^{2} + 12 \times {3 + x} = 85

Operations and Ordering - Key takeaways

It is important to evaluate numerical and algebraic expressions in a specific order to guarantee a correct answer.
There are 4 steps to follow to ensure you are calculating your operations in the right order;
- Grouped terms, parentheses
- Powers or exponents
- Multiplication or division (in order from left to right)
- Addition or subtraction (in order from left to right)
There is an acronym that can be used to help you remember the correct order of operations, it is PEMDAS.
It is also important to follow the correct order of operations when plugging in a given value for the variable in an equation.

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Frequently Asked Questions about Order of Operations

What is order of operations?

Order of operations is a correct order in which you should solve operations, operations are things such as addition, multiplication, and powers.

What are the rules of order operations?

The rules for order of operations state that you should complete a sum in the following order;

Calculate anything inside grouping symbols
Calculate any powers
Calculate any multiplication or division
Calculate any addition or subtraction

What are the importance of order of operation?

It is important to use order of operation to get the correct answer to your sum, if you simply calculate the sum from left to right you will be left with the wrong answer.

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