Roots and Powers of Algebraic Expressions

Grouping Symbols in Math: Definition & Equations

This lesson will investigate grouping symbols in math, which allow steps in a math problem to receive priority attention. Incorrect use of grouping symbols can make the difference between obtaining the right or wrong answer.

Overview of Grouping Symbols

A math problem can contain many different operations. Whenever numbers or variables and a math operation are contained within grouping symbols, it is like that part of the problem is saying, 'Do me first!' The grouping symbols most commonly seen in mathematical problems are parentheses, brackets, and braces. In a math problem, all three serve the same purpose--to make sure that whatever is contained within those symbols gets attention first.

Evaluating Expressions

To demonstrate the difference grouping symbols can make in a calculation, consider the problem 2(3) + 7. Without grouping symbols, the order of operations says multiply first, and then add. This order would produce the answer 6 + 7 = 13. However, if I were to add a grouping symbol and make the problem instead 2(3 + 7), then first attention must be given to the parentheses. This order would produce the answer 2 (10) = 20.

Notice that even though no multiplication symbol was shown, multiplication between the 2 and the 10 was implied. This is true for all grouping symbols. If no operation is shown between a number and a grouping symbol, it means multiply.

Grouping symbols can help clarify written or typed mathematical expressions. For example, if you type the expression 12-4/2, does this mean subtract 4 from 12 then divide by 2, or does it mean divide 4 by 2 and subtract that from 12? The answer will be different depending on which way the expression is evaluated. To make the meaning clear, you could put grouping symbols around the operation you want to be done first. (12 - 4)/2 indicates the subtraction should be performed first. 12 - (4/2) indicates the division should be performed first.

Some problems contain more than one set of grouping symbols. Examples are problems such as (5 + 2)/(8 - 1) or {2 + (6/2)}.

In the first problem, each grouping symbol receives separate but equal priority.

• (5 + 2)/(8 - 1) = 7/7 = 1.

In the second problem, the innermost grouping symbol receives first priority. In other words, if there are grouping symbols contained within grouping symbols, start the problem from the innermost grouping symbol first and work your way outward.

• {2 + (6/2)} = {2 + 3} = 5

It is important not to drop the grouping symbols too soon. For example, consider the following problem:

• 18 + {2 + 3 (6 - 1*2)}

Since there are 2 operations in the parentheses, the ( ) must not be dropped until you completely simplify the expression within.
• 18 + {2 + 3 (6 -1*2)}
= 18 + {2 + 3(6 - 2)}
= 18 + {2 + 3(4)}
= 18 + {2 + 12}
= 18 + {14}
= 18 + 14
= 32}

Expressions with Variables

Sometimes, a grouping symbol contains an expression with both numbers and variables. Since these are unlike terms, you cannot combine them as you would with something like (6 - 1*2). The distributive property in mathematics gives us a way to eliminate the grouping symbols in cases like this. For example, in the problem 3(x + 5), you cannot combine the x and the 5. However, using the distributive property, you could multiply both of these terms by the factor 3 outside the parentheses and in doing so eliminate the grouping symbols.

• 3(x + 5) = 3*x + 3*5 = 3x + 15

If there is no factor outside the grouping symbol and no operation other than addition, you can safely remove the symbols and combine any like terms.

A situation that is commonly encountered is having a negative sign immediately outside of the grouping symbol. In all cases, the negative sign must be applied to whatever is in the grouping symbol when the grouping symbol is removed.

Consider the problem - (4*3 + 6). After simplifying the expression in the parentheses, you must apply the negative sign as soon as the parentheses are removed.

• - (4*3 + 6)= - (12 + 6) = - (18) = - 18

If you use the distributive property to remove parentheses preceded by a negative sign, you must apply the negative sign outside the parentheses to each term in the parentheses.

• - (2x + 3y - 7) = - 2x - 3y + 7

Sometimes, you can also use the distributive property to make a problem easier. An example is if you want to multiply 24*99
1. Rewrite 99 as 100 - 1, use this expression in parentheses and apply the distributive property
24*(100 - 1) = 24*100 - 24*1 = 2400 - 24 = 2376
This method works well with this problem because the number 99 is close to 100 and it is usually easier to calculate
with a power of 10. .

Equations

When grouping symbols occur in equations, the distributive property must be applied to remove them. To solve the equation 4 + 5(x - 1) = 5x - (x + 3), the steps are as follows:

Lesson Summary

The most common grouping symbols used in mathematics are parentheses, brackets and braces. These grouping symbols in math provide a way to indicate that the expressions contained within them are evaluated first. When removing grouping symbols, it is important to apply any negative signs outside the grouping symbol. For expressions that cannot be simplified and are contained in a grouping symbol, the distributive property allows us to remove the grouping symbol using multiplication.