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# Parentheses|Definition & Meaning

**Definition**

In mathematics, **parentheses** ( ) are used to enclose numbers or variables that are part of a mathematical expression. They can be used to indicate the order in which operations should be performed, or to group terms.

It is common for math problems to contain a greater number of symbols and expressions as the level of difficulty in the problem increases. Some symbols are easy to recognize in the beginning, such as the main **operation** **sign**, **equals** **sign**, and **greater** **than** and **less** **than** symbols. It is important to keep in mind that parentheses should not be confused with either **braces** or **brackets**.

Figure 1 – Symbol of Parentheses

**Parentheses and Order of Operation **

There is a concept called the **order of operations** that is accompanied by **parentheses** that indicates that whatever is in the parenthesis must be solved first when it comes to the order of operations.

When multiple **operations** are part of a single problem, the order of operations specifies which should be done first. People typically refer to it as **PEMDAS**, which stands for the pneumonic device.

**Use of Parentheses in Maths**

**Parentheses** can be used to separate numbers in math. Math problems sometimes contain negative numbers inside parentheses.

For example,

**7 + (-9) = -2. **

An operation involving a **negative** number is done first, just like a separate problem, since it is part of the operation. When the number is negative by itself, it will be written that way so as not to confuse the minus sign with the negative sign.

- Without parentheses
**12 – – 6 = 18** - With parentheses
**12 – (-6) = 18**

The **negative** **number** is more obvious when it is present within parentheses. Whenever this occurs, we have to solve the problem using **PEMDAS,** and we have to pretend that the equation in the parentheses has already been solved. The following example shows a parenthesis problem with negative numbers.

**20 + (-3 × 4 + 6)**

Steps | Operation | Results | Explanation |

1 | -3 × 4 = -12 | 20 + (-12 + 6) = | The first is to multiply the equation inside the parentheses. i.e., -3×4=-12 |

2 | -12 + 6 = -6 | 20 + (-6) = | Consider the negative -12 as part of the addition problem. |

3 | 20 + (-6) =14 | 16 | The number is still negative, so you can leave the parentheses out of the equation and solve it. |

The **parentheses** are often used to prevent confusion when two negative numbers are multiplied by one another. In such a case, the problem will look like this (-6) (-8).

**Use of Parentheses for Multiplication**

In math, **parentheses** are used for **multiplying** numbers. Parentheses indicate that you have to multiply an equation if there is no arithmetic operation. To better understand this, let’s take a look at an example

**3 (2 + 2) = 3 × 4 = 12**

To show two parentheses being multiplied, there are two possible ways to do so. There are two options to separate the two parentheses: either by placing a **multiplication** **symbol** between them or by placing them **directly** next to one another. Consider the following example (9/3) (12-9)

Step | Operation | Result | Explanation |

1 | 9/3 = 3 | 3(12 – 9) = | In this case, since there are two sets of parentheses, we should start with the left parenthesis. |

2 | 12 – 9 = 3 | 3 × 3= | Once the operations within the parentheses have been completed, the parentheses can be removed |

3 | 3 × 3 = 9 | 9 | Once both operations are completed, we can multiply the results. |

Figure 2 – Some basic uses of Parentheses

**Use of Parentheses in Different Mathematical Terms**

- In math,
**parentheses**are used to group numbers and specify how operations are performed. - Parentheses can be used to enclose a
**function’s****variables**in the form, such as f(x), so that the values of the function will depend on the values of its parameters. - A binomial
**coefficient**is denoted by large parentheses around two numbers. - In a set of two or more numbers, such as (a, b, c), parentheses around them are meant to indicate that there is an n-tuple of numbers connected in some kind of special way.
- The greatest
**common****divisor**can also be indicated by using parentheses.

For example, (6, 22) GCD of 6 and 22 is 2.

## The Order of Operations

The **order** of operations is changed by parentheses.

When there are multiple symbols in an equation, follow this order

- Parentheses
- Exponents
- Multiplication/division (left to right)
- Addition/subtraction (left to right)

You should look at the terms present in the parentheses of an equation before you begin to attempt to solve it.

To better understand this, let us take a look at an example.

**8 – 12 ÷ 4 – 2 × 1 + 6**

We can solve this by applying the order of operations we learned earlier.

**= 8 – 12 ÷ 4 – 2 × 1 + 6**

**= 8 – 3 – 2 × 1 + 6** (The first step is to divide)

**= 8 – 3 – 2 + 6** (The second step is to multiply)

**= 5 – 2 + 6**(The third step is to subtract)

**= 3 + 6** (again subtract)

**= 9** (The fourth step is the addition)

Have you noticed? Due to the presence of **parentheses** in the equation, the answer to the same equation changed.

You should solve the inner expression of a **parenthesis** first if there are parentheses inside parentheses.

Here is an example that will help us better understand this

Simplify the following expression **(3 + (5 × 4))**

The inner bracket will be solved first, so the expression becomes **3 + 20 = 23.**

**Solved Examples**

Figure 3 – Steps to solving problems

**Example 1**

**Simplify the following expression: (4 + 6 × 8) – 6 + (3 × 4)**

### Solution

Solve the expressions within the parentheses first.

**= (4 + 48) – 6 + 12** (multiply the numbers inside the parentheses)

**= 52 – 6 +12** (solve the terms that are present inside the parentheses)

**= 46 + 12** (there are no more parentheses; subtract 6 from 52)

**= 58 **(add the two terms together)

**Example 2**

**Simplify the following expression: (3 × (8 – 6)) – ((8 ÷ 4) + 2)**

### Solution

Innermost parentheses are solved first, so we perform 8 – 6 and 8 **÷ **4, both of which result in 2:

**= (3 × 2) – (2 + 2) **

**= 6 – 4 **(solve the parentheses terms)

**= 2 **(subtract the resulting terms at the end)

**Example 3**

**Simplify the following expression: 3(4 + 6) + 9(3 – 1)**

### Solution

First of all, we will solve the expression that is present inside the parenthesis

It is important to note that the parentheses also indicate a **multiplication sign** in this case. Whenever a number outside parentheses is written next to it without an operator on either the left or right side, it shows an implicit multiplication.

**= 3 × 10 + 9 × 2** (solve terms inside parentheses and write out implicit multiplication with the results)

**= 30 + 18 **(solve multiplication first as per BODMAS)

**= 48 **(finally add the two terms)

**Example 4**

**Solve the expression: 5(2 + 6) + 5(10 – 2)**

### Solution

First of all, we will solve the expression that is present inside the parenthesis

**= 5(8) + 5(8)**

**= 40 + 40**

**= 80**

*All the mathematical images are generated using GeoGebra.*