Multiple grouping symbols are used to organize algebraic problems. Parentheses (), brackets [ ], and braces { } are examples of relational symbols that indicate the beginning and end of a group, and aid in determining the order in which arithmetic operations should be performed. Keep reading to discover what grouping symbols mean, the types of grouping symbols as well as the different rules for orders of operations.
What do grouping symbols mean in math?
Grouping symbols are often used to indicate that a certain group of numbers and/or operations should be treated as one whole entity.
Here are common grouping symbols used in mathematisc:
- Parentheses (): The most popular grouping symbol is the parenthesis.
- Brackets [ ] and braces { }: Similar to parentheses, brackets and braces are widely used to group variables together. When there are multiple groupings in a math problem, using the various symbol types is helpful.
- The radical symbol √ : This is often known as the square root symbol, and is used to find roots.
- The fraction line: This is also known as a vinculum and serves as a grouping symbol. In the numerator, everything that is above the line is together, while in the denominator, everything below the line is together.
Before anything outside the grouping sign may act on the terms inside it, those terms must be operated upon. Each bracket type is equally important. None is more potent or behaves in a way that is different from the others.
What are the grouping symbols in the order of operations?
There are guidelines to follow while evaluating a mathematical expression made up of several brackets. This is referred to as the brackets’ rules of operation or order of operation. The innermost bracket’s values must be simplified first in a particular situation, according to the general sequence of action of the bracket, which is depicted as [ { () } ].
This indicates that the () brackets will be solved first, followed by the { } brackets, and finally the [ ] brackets. Finding an exponent and solving it first if there is one is the second stage to solve a problem. In the third phase, we search for expressions that have division or multiplication operators. We check the arithmetic expression from left to right to see if both operators are present. We then start by solving the operator that comes before it.
Why is it important to follow the order of operations?
Type of brackets
In mathematics, three different types of brackets are typically used:
- Circle brackets or parentheses: ( )
- Brace or curly brackets: { }
- Box or square brackets: [ ]
Parentheses
Also known as round brackets and expressed as ( ), these are the brackets that are used most frequently. They are utilized to combine several numbers and formulae. Simply putting round brackets around numbers is the same as using a multiplication symbol.
For instance, (3)(4)=12
Negative numbers can also be expressed mathematically using them.
For instance, 5 + (-4) = 1, rather than 5 + – 4 = 1
Numbers and their exponents can be separated using parentheses. For instance:
(2 + 4), 5(111), and 25 – (12 + 8).
Braces
Curly brackets are used to arrange distinct mathematical notions similar to parentheses but they can also be used to represent normal subgroups, sets, or to create nested expressions. For example, [4 + [3 (- 2)] – [{(4 6) + (14 7)} – (- 3)]
Square brackets
To distinguish between sub-expressions of a complicated mathematical expression, square brackets are typically used. Examples include 10 x [(4 – 2) x (4 x 2)] and [100 – (3 – 1) + (7 x 8)].
Why do we use grouping symbols?
Consider the calculation 2(3) + 7 to see the impact grouping symbols can have on a calculation. The order of operations without grouping symbols is multiplication followed by addition. The result of this would be 6 + 7 = 13. However, if we were to add a grouping symbol and change the problem to 2(3 + 7), then the parentheses would require the most focus. The result of this arrangement would be 2 (10) = 20.
Even though a sign for multiplication was not displayed, multiplication between the numbers 2 and 10 was suggested. For all grouping symbols, this is true. Multiplication is indicated if there is no operation symbol displayed between a number and a grouping symbol. Mathematical expressions can be made clearer by grouping symbols.
When you write the expression 12 – 4/2, for instance, what does it actually imply? Does it mean that you are dividing 4 by 2 and then subtracting that from 12? Depending on how the expression is evaluated, the response will vary. In order to make the message clear, it is important to put grouping symbols around the operation you want to do first. For example, (12 – 4)/2 suggests that subtraction comes first, whereas 12 – (4/2) suggests that division should be done first.
Rules for orders of operations
Observe the expression
It is important to take note of the expression. To begin with, we solve the equation within the brackets. We then resolve grouping operations from the inside out. There is a certain way to solve the parenthesis, i.e., [ { () } ]. Therefore, take note of the pattern of brackets that are present in the phrase. Start by resolving the round () and curly {} and box [] brackets. The order of operations is to be followed in the parentheses.
Exponents
Search for and solve any term that is present in the form of exponents, after solving the numbers in the parenthesis.
Multiply and divide
We are now down to the fundamental four operators: addition, subtraction, multiplication, and division. If there are any equations with multiplication or division from left to right within them, it is important to solve them.
Add and subtract
Lastly, solve the equations with addition and subtraction from left to right. The acronyms PEMDAS, or BIDMAS, or BODMAS are used to refer to this practice.
How do you solve grouping symbols?
Question 1
18 + {2 + 3 (6 – 1 x 2)}
The () cannot be removed until the expression inside the parenthesis { } has been entirely simplified, because there are two operations there.
18 + {2 + 3 (6 -1 x 2) }
We take the inner bracket (6 – 1 x 2) and, according to BODMAS, we begin with the multiplication:
= (6 – 2)
Then, we subtract:
6 – 2 = 4
Returning to the original question, we then insert our answer as follows:
= 18 + {2 + 3(4)}
Now we solve the following bracket { }:
{2 + 3(4)}
If you are unsure, 3(4) is the same as 3 x 4, therefore:
2 + 3 x 4
We multiply:
2 + 12
And solve:
2 + 12 = 14
We then insert this answer into our equation:
= 18 + {14}
And solve the remaining calculations:
= 18 + 14
= 32
That’s the correct answer. It’s that easy!
Question 2:
[ 3 + { 30 – 3(4 x 2) ] + 12 + {12 + (17 – 3 x 2) } = ?
Now this is a complex one. Let us begin by treating it as 3 separate problems that we have to solve, before tackling the whole equation:
[ 3 + { 30 – 3(4 x 2) } ] + 12 + {12 + (17 – 3 x 2) } = ?
To make things easier, let us work from left to right and start with the red equation:
[ 3 + { 30 – 3(4 x 2) } ]
According to the order of operations, the innermost bracket’s values must be simplified first. Following which, you can work your way out according to: [ { () } ]. Let us begin as follows:
3(4 x 2)
First we multiply within the bracket:
4 x 2 = 8, resulting in 3(8)
3(8) is the same as 3 x 8, so let us now solve this to get 24.
We then insert this into our next equation within the { } brackets:
{ 30 – 24 }
Subtracting these numbers gives us:
{ 6 }, which we can then insert into the next bracket equation [ ]:
[ 3 + {6} ]
This equals 9. Let us insert this into our original equation:
[ 9 ] + 12 + {12 + (17 – 3 x 2) } = ?
Our red equation is now simplified, and the blue equation cannot be simplified any further. Let us therefore move on to green:
{12 + (17 – 3 x 2) }
Firstly, we solve the innermost bracket ( ):
(17 – 3 x 2)
According to BODMAS, we should begin by multiplying 3 x 2, and then use this answer and subtract it from 17:
3 x 2 = 6, therefore 17 – 6 = 11
Inserting this in our equation renders:
{12 + (11) }
Then, we solve 12 + 11, which equals 23, and insert it in our original equation:
[9] + 12 + {23} = ?
As all our brackets have been simplified, we can then solve the problem, by adding them together to get 44:
[ 3 + { 30 – 3(4 x 2) ] + 12 + {12 + (17 – 3 x 2) } = 44
Math tutoring services
Many students find math difficult, and once you fall behind, it can be difficult to catch up. Fortunately, Tutorax offers online and in-home tutoring services for learners of all ages. Our tutors can assist you with homework, test preparation, and classroom assistance. If your child is struggling with math, tutoring can go a long way towards improving their academic performance and motivation.