**Definition**

An expression is a sentence with a minimum of two numbers and at least one math operation. An algebraic expression (or) a variable expression is a combination of terms by the operations such as addition, subtraction, multiplication, division, etc. an extension to the algebraic expressions are the polynomials. Also, an algebraic expression, in which variable(s) does (do) not occur in the denominator, exponents of variable(s) are whole numbers and numerical coefficients of various terms are real numbers, is called a polynomial.

Let us understand it through some examples –

Consider the following statements –

- A number x increased by 7
- 9 exceeds a number x
- 4 times a number x
- A number y divided by 5
- The sum of a number x and twice the number y

We can write the above expressions as under –

- x + 7
- 9 – x
- 4x
- $\frac{y}{5}$
- x + 2y

**Using Grouping Symbols to Write Expressions**

In simplifying mathematical expressions consisting of the same type of operation, we perform one operation at a time generally starring from the left towards the right. If an expression has more than one fundamental operation, you cannot perform operations in the order they appear. Some operations have to be performed before than the others. This is each operation has its own precedence. Generally, the order in which we perform operations sequentially from left to right is division, multiplication, addition, subtraction. But when the expressions make use of brackets as well, we have a set of rules that defines the precedence of the operations. Let us learn the manner in which this rule is defined.

Let us understand this through an example.

Suppose we want to find the value of 2 + 3 x 5

There are two ways to find the value of the above expression.

In the first method, let us move from left to right, solving the expression in that order. This means, first we will find the sum of 2 and 3. We will get, 2 + 3 = 5.

Now we have 2 + 3 x 5 = 5 x 5

Now, we find the product of 5 with 5 to get the answer as 25. Therefore, we have,

2 + 3 x 5 = 25 ……………………………….. ( 1 )

Now, let us find the value of this expression through another method.

We will first find the value of 3 x 5 and add this product to 2. So, we get,

3 x 5 = 15 and 15 + 2 = 17

Hence, now we have,

2 + 3 x 5 = 17 ………………………… ( 2 )

From ( 1 ) and ( 2 ) we can see that both the results obtained are not the same. This validates the need for having precedence of operators so as to receive the same answer to the given algebraic expression.

**How to Solve Expressions involving Grouping Symbols?**

Simple expressions which involve one of more than one operator can be solved using BODMAS. What is BODMAS and how it is used to solve simple expressions? Let us find out.

Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations. Without consistent order of operation, one can make big mistakes during computation. There are a number of rules that define the order of operations, depending upon the involvement of the operators, brackets exponents and other mathematical symbols for operations. This rule is known as BODMAS, where

B stands for Brackets

O stands for Of

D stands for Division ( ÷ )

M stands for Multiplication ( x )

A stands for Addition ( + )

S stands for subtraction ( – )

**BODMAS**

Understanding BODAMS is integral to the understanding of the use of grouping symbols in expressions. This is because it is the BODMAS rule that defines the order in which the operations are to be performed in the case of mathematical expressions. In other words, it defines the order in which operations on numbers are to be performed when there are situations where we have two more than two operations in an expression.

Let us consider an example. Suppose we want to find the value of 3 + 5 x 2.

Here, we have two operators, namely ( x ) and ( + ). We know that in mathematics, we can perform operations between two numbers only in one go. If we have more than 2 numbers, the result from the first operation is used for performing the next operation. So, in the above example, there are two operations to be performed, ( 3 + 5 ) and ( 5 x 2 ). We need to decide, which one to start with. This is where the MDAS rule comes into play.

From this rule, we can see that Multiplication precedes addition. Therefore, we will perform the operation multiplication first. The result thus obtained will be added to 3. The steps involved in this process will be –

3 + 5 x 2

= 3 + 10

= 13

**Hence, 3 + 5 x 2 = 13**

Now, that we have understood what we mean by grouping symbols and mathematical expressions let us learn about the steps involved in solving mathematical expressions involving grouping symbols. The steps involved are –

- Always start by calculating all expressions within parentheses

The first thing is that we should look for removing any grouping symbols in the algebraic expression. This means that at the top of the list, remember to always** **simplify everything inside the grouping symbols. Examples of grouping symbols are parentheses ( ), brackets, and braces { }. For nested grouping symbols, work it out from the inside and out.

- Simplify all the exponents such as square roots, squares, cube, and cube roots

After having removed the parenthesis, we come to the next step of solving all exponential values in the algebraic expression. Exponential expressions which may include root values such as square roots, squares, cube, and cube roots etc. are calculated or evaluated first before performing any of the four fundamental arithmetic operations, namely: addition, subtraction, multiplication, and division.

- Perform the multiplication and the division starting from left to right

Next, multiply and/or divide whichever comes first from left to right before performing addition and subtraction. This tells us that multiplication and division have a higher level of importance than addition and subtraction.

- Finally, do the addition and subtraction similarly, starting from left to right.

Let us understand it using an example.

**Example** Simplify 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

**Solution** We have been given the expression

95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – $\overline{17-10}$ } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

95 – [ 144 ÷ ( 12 x 12 ) – ( – 4 ) – { 3 – 7 } ]

Next, we will remove the innermost bracket to get,

95 – [ 144 ÷ 144 + 4 – { 3 – 7 } ]

Now, we will remove the curly braces to get

95 – [ 144 ÷ 144 + 4 + 4 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

95 – [ 1 + 4 + 4 ]

Next, we will remove the square brackets to get,

95 – 9

= 86

**Hence, simplification of 95 – [ 144 ÷ ( 12 x 12 ) – ( -4 ) – { 3 – **$\overline{17-10}$** } ] = 86**

**Converting Simple Expression to Equations**

We know that a statement involving the symbol “=” is called a **statement of equality** or simply an **equality**. A simple expression can be converted into an equation when we have both the left hand side and the right-hand side of the expression separated by a comparative sign such as an equal sign. Let us consider the expressions we discussed above and convert them into equations.

The simple expressions we had were as below –

- A number x increased by 7
- 9 exceeds a number x
- 4 times a number x
- A number y divided by 5
- The sum of a number x and twice the number y

We can write the above expressions as under –

- x + 7
- 9 – x
- 4x
- $\frac{y}{5}$
- x + 2y

Now, if these were to be converted into equations, they could have been written in the following manner –

Statements –

- A number x increased by 7 is 15.
- 9 exceeds a number x by 3
- 4 times a number x is 24
- A number y divided by 5 is 7
- The sum of a number x and twice the number y is 12

We can write the above statements as under –

- x + 7 = 15
- 9 – x = 3
- 4x = 24
- $\frac{y}{5}$ = 7
- x + 2y = 12

In this manner, simple expressions can be converted into equations.

**Classification of Simple Expressions**

Simple algebraic expressions can be classified as monomials, binomials etc. depending upon the number of variables represent in the expression. Let us recall that Polynomials are algebraic expressions that consist of variables and coefficients. In other words, an algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

**Monomial** – An expression having a single term with non-negative exponential integers is called a monomial. For example, 2 x is a monomial.

**Binomial** – An expression having two terms with non-negative exponential integers is called a binomial. In other words, an expression formed by the addition or subtraction of two monomials is called a binomial. For example, 2 x + 5 y is a binomial.

**Trinomial** – An expression having three terms with non-negative exponential integers is called a trinomial. In other words, an expression formed by the addition or subtraction of three monomials is called a trinomial. For example, 2 x + 5 y + 6 z is a trinomial.

**Simple Rational Expressions**

Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. For example, $\frac{x}{x-2}$ is a rational expression as it has polynomials in both its numerator as well as its denominator.

**Simplifying Rational Expressions**

Simplification of a rational expression is the process of reducing a rational expression in its lowest terms possible. It is similar to reducing fractions or rational numbers to their lowest form. The following steps are sued for simplification of rational expressions-

- Factorize both the denominator and numerator of the rational expression. Also, remember to write each expression in standard form.
- Reduce the expression by cancelling out common factors in the numerator and denominator
- Rewrite the remaining factors in the numerator and denominator.

Let us understand the above steps by an example.

**Example**

Simplify the rational expression $\frac{3x^2- 3x}{3x^3- 6x^2+3x}$

**Solution**

We are given the rational expression

$\frac{3x^2- 3x}{3x^3- 6x^2+3x}$

In order to simplify this rational expression, first, we will look for factors that are common to the numerator & denominator. We have, 3x is a common factor the numerator & denominator. Note that it is clear that x ≠0. Hence we will cancel the common factor to get,

$\frac{3x ( x -1 )}{3x ( x^2-2x+1 )}$

Now, if possible, we will look for other factors that are common to the numerator and denominator. We can see that ( x – 1 ) is also a common factor of the numerator and the denominator as x^{2}-2x+1 can be written as ( x – 1 ) ( x – 1 ).

Therefore, we now have,

$\frac{( x -1 )}{( x-1 )( x-1 )}$

We now have,

$\frac{1}{( x-1 )}$

**Therefore, the rational expression **$\frac{3x^2- 3x}{3x^3- 6x^2+3x}$** in its simplified form will be **$\frac{1}{( x-1 )}$

**Solved Examples**

**Example 1** Simplify the expression 197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ]

**Solution** We have been given the algebraic expression,

197 – [1/9 { 42 + (56 – $\overline{8+9}$ ) } +108 ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression; hence it needs to be solved first.

Solving the vinculum, we will get,

197 – [1/9 { 42 + (56 – 17 ) } +108 ]

Next, we will remove the innermost bracket to get,

197 – [1/9 { 42 + 39 } +108 ]

Now, we will remove the curly braces to get

197 – [81 / 9 +108 ]

Now, it is important to see that within the square bracket we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

197 – [9 + 108 ]

Next, we will remove the square brackets to get,

197 – 117

Last, we just need to find the difference of the remaining two values to get,

80.

**Hence, simplification of 197 – [1/9 { 42 + (56 – **$\overline{8+9}$** ) } +108 ] = 80.**

**Example 2** Simplify the expression 15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

**Solution** We have been given the expression

15 – ( – 5) { 4 – $\overline{7-3}$ } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

We will use PEMDAS to solve the above expression.

Note, we can see the presence of vinculum in the expression, hence it needs to be solved first.

Solving the vinculum, we will get,

15 – ( – 5) { 4 – 4 } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ]

Next, we will remove the innermost bracket to get,

15 + 5 x 0 ÷ [ 3 { 5 + 18 } ]

Now, we will remove the curly braces to get

15 + 0 ÷ [ 3 x 23 ]

Next, we will remove the square brackets to get,

15 + 0 ÷ 69

Now, it is important to see that we have two operations to be performed, division and addition. So, going by the PEMDAS rule, we will first perform division to get,

15 + 0

= 15

**Hence, simplification of 15 – ( – 5) { 4 – **$\overline{7-3}$** } ÷ [ 3 { 5 + ( -3 ) x ( -6 ) } ] = 15**

**Key Facts and Summary**

- An expression is a sentence with a minimum of two numbers and at least one math operation.
- An algebraic expression (or) a variable expression is a combination of terms by the operations such as addition, subtraction, multiplication, division, etc. an extension to the algebraic expressions are the polynomials.
- Order of operation can be defined as a standard procedure that guides you on which calculations to begin within an expression with several arithmetic operations.
- BODMAS, stands for B stands for Brackets, O stands for Of, D stands for Division ( ÷ ), M stands for Multiplication ( x ), A stands for Addition ( + ), S stands for subtraction ( – ).
- A statement of equality that involves one or more variables is called an equation.
- An expression having a single term with non-negative exponential integers is called a monomial.
- An expression having two terms with non-negative exponential integers is called a binomial.
- An expression having three terms with non-negative exponential integers is called a trinomial.
- Rational expressions are fractions that have a polynomial in the numerator, denominator, or both.

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