**8.1 Introduction**: Expression is a term given to any mathematical symbol. In algebra we use both, numbers and letters. Algebraic rules are similar to the arithmetic rules; the difference is we use letters to represent a general rule or general formula using letters, assuming it to be true for any number.

For example: in arithmetic 7 + 3 = 3 + 7

In algebra a + b = b + a

Here a represents the first number and b represents the second number. a and b can be any number, satisfying the mentioned law.

An algebraic expression is made up of signs and symbols of algebra. These symbols include Arabic numeral, literal numbers etc. The numbers are called the numerical symbols, and the letters are called literal symbols.

i) Operation – Addition

Sign – “+”

a + b

In algebra we call it “sum” of two numbers.

ii) Operation – Subtraction

Sign – “-“

In algebra we call it as “difference” between numbers.

iii) Operation – Multiplication

Sign – “x” or “**.**” or “*”

a x b or a.b or a* b or simply ab

In algebra we call it as “product of numbers.

iv) Operation – Division

Sign – “/”

In algebra we call it as a “quotient”.

When numbers are added or subtracted they are referred as **Terms** and when numbers are multiplied they are called **Factors**.s

In algebra we speak “**sum**” for of several **terms** even if there are subtractions in it.

That is, find the **sum** of the four **terms**

a -b + c – d

Although the expression above has subtraction in it yet we call it as **sum** of the given **term.**

When a number is multiplied by itself repeatedly then we get the **Power** of that number.

**a.a** is called the **second power** of **a**, **a.a.a** is called the **third power** of **a** and so on.

In general **a.a.a……..n times** is called the **nth power of a**

Now,

a.a = a^{2}

a.a.a = a^{3}

a.a.a.a = a^{4}

a.a. to n times = a^{n} (where n can be any number)

a^{2} is read as “a squared” and a^{3} is read as “a cubed”. The small numbers placed on a are called **Exponents**. In a^{2}, **2** is the exponent, similarly in a^{3}, **3** is the exponent

Parentheses signify that what they enclose should be treated as one number.

4 + (4 x 3) = 4 +12

5(7 + 2) = 5 x 9

9 + (1+ 6) = 9 + 7

10(3 x 2) = 10 x 6

**Remember that if there is no sign between a number and parentheses it means multiplication**

Order of operation generally includes the BODMAS rule

That is **B**racket or parentheses **O**pen then **D**ivision then **M**ultiplication and then **S**ubtraction

An algebraic expression should be solved by following the above mentioned sequence.

Order of operation

Step1. Evaluate parentheses, in the expression first if there are any.

Step2. Evaluate powers or exponents.

Step3. Perform Division

Step4. Perform Multiplication

Step5. Perform Addition

Step6. Perform Subtraction

Let’s solve an expression to understand the rule better.

Solve 4+6(1+2)2-3 x 6/2

Step1.Evaluating parentheses

4 + 6 (3)2 – 3/6

Step2. Evaluating exponents or power

4 + 6 x 9 – 3/6

Step3. Performing division

4 + 6 x 9 – 2

Step4. Evaluate the product

4 + 54 – 2

Step5. Evaluate the sum

58 – 2

Step6. Evaluate the difference

56

The solution is 56

Taking another example with no parentheses and exponents

Solve 12 – 3 x 6 + 14/7

Start with division (Rule3 since there is no use of rule 1 and 2)

12 – 3 x 6 +2

Now the next step is multiplication

12 – 18 + 2

Now add

12- 20

Now subtract

-8 ans.

Although there is no such hard and fast rule between addition and subtraction, their order of operation can be reversed.

Evaluation of expression means replacing the letter with its value followed by the order of operation.

Example: if *x* = 5, y = 10 and z = 2

Solve i) *x *+ (y x z)

= 5 + (10 x 2)

= 5 + 20

= 25

ii) *x* – (y + z)

= 5 – (10 + 2)

= 5 – 12

= -7

iii) y(*x*/z)

= 3(10/2)

= 3 x 5

= 15

iv) *x*2 + y3 + z2

= 52 + 103 + 22

= 25 + 1000 + 4

= 1029

A variable is a literal symbol and is so called because its value may vary.

A variable, like *x*, is a blank or empty symbol. And we can assign any value, whether a positive number or a negative number, a whole number or a fraction.

**Algebraic expression with two variables:** In these expressions value of one variable depends on the value of other variable

y = b*x*+c

Where b and c are constants, while *x* and y are variables.

Example y = 4*x* + 6

Here value of y depends on *x*

When *x* = 0, *y* = 4**·** 0 + 6 = 0 +6 = 6

When *x* = 1, *y* = 4**·** 1 + 6 = 4 + 6= 10

When *x* = 2, *y* = 4**·** 2 + 6 = 8 + 6 = 14

When *x* = 3, *y* = 4**·** 3 + 6 = 12 + 6 = 18

When *x* = 4**, y = 4· 4**

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