**Variables:** Variables are the alphabets or letters used to represent unknown numbers/unknown quantities.

- These are also known as literals.

**For ****example:** \(x,\;y,\;a,\;m\) etc. can be used as variables.

- A variable can be used in any sort of mathematical expression.

- An expression having numbers connected by one or more operations, with no equals sign, is a numerical expression.

**Example:** \(2(8+6)\)

- An expression having variable(s) with one or more operations along with numbers with no equals sign, is an algebraic expression/variable expression.

**Example:** \(6x+7,\;3y-3,\;2z\) etc.

A \(2(8+7)\)

B \(7(3\div2)\)

C \(3(7)+2(4)\)

D \(3x+2t\)

- Parentheses are used to separate a part of any numerical expression.

For example: \(\dfrac{22}{7} × (42 \div 7-3) - 1\)

Here, \(42 \div 7 -3\) is separated by parentheses.

- The position of the grouping symbols (parentheses) can change the value of a numerical expression.

For example: \(5 + 8 × 2 \div 4\)

Without parentheses, the solution would be:

\(=5 + 8 × 2 \div 4\)

\( = 5 + 16 \div 4\)

\( = 5 + 4\)

\( = 9\)

Using parentheses at \((5 + 8)\), the solution would be:

\( = (5 + 8) × 2 \div 4\)

\( = 13 × 2 \div 4\)

\(=26 \div 4\)

\( = \dfrac{13}{2}\)

Using parentheses at \((5 + 8 × 2)\), the solution would be:

\(=5 + 8 × 2 \div 4\)

\( = (5 + 8 × 2 ) \div 4\)

\(= (5 + 16) \div 4\)

\(=21 \div 4\)

\(= \dfrac{21}{4}\)

- It can be observed that by altering the position of parentheses, the value of the expression also gets changed.

A \((10 0\div 5)× 6 - 2\)

B \(100 \div (5 × 6 )- 2\)

C \(100 \div (5 × 6 - 2)\)

D \(100 \div 5 × (6 - 2)\)

- PEMDAS rule is an order to evaluate any numerical expression containing two or more than two operations.
- PEMDAS rule is as follows:

(i) P stands for parentheses ( )

(ii) E stands for exponents ^

(iii) M stands for multiplication ×

(iv) D stands for division \(\div\)

(v) A stands for addition +

(vi) S stands for subtraction –

It is clear from above that the parentheses are to be evaluated first, next the exponents, then the multiplication or the division in the order from left to right and at last the addition or the subtraction in the order from left to right.

- In any expression, if one of the operations is missing then we will evaluate the next one.
- PEMDAS rule can be remembered easily by the sentence.

"**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally"

- Consider the following example: \(3^3× 1 - (8 \div 2) + 4\)

Here, all the operations are used, thus, we shall follow the PEMDAS rule to evaluate.

\(3^3 × 1 - (8 \div 2) + 4\)

(i) Here, only one pair of parenthesis is used so, we shall evaluate it first.

\(8 \div 2 = 4\)

(ii) Now, the expression is \(3^3 × 1 - 4 + 4\)

Now, the exponent is to be evaluated.

\( = 27 × 1 - 4 + 4\)

(iii) Multiplication is to be done.

\( = 27 - 4 + 4\)

(iv) Subtraction is to be done.

\(=23+4\)

(v) Addition is to be done.

\(= 27\)

**Note**: By not following the PEMDAS rule, we would get different and incorrect answers.

- Algebraic expressions are classified on the basis of number of terms.

Various types of algebraic expressions are as follows:

**Monomial:**An expression which has only one term is called monomial.

Eg. \(4x,\; 3y,\; 6z\), etc.

**Binomial:**An expression which has two terms is called binomial.

Eg. \(x^2+4x,\;y+4,\;3x+4\), etc.

**Trinomial:**An expression which has three terms is called trinomial.

Eg. \(x^3+2x+1,\;a^2+b^2+2ab\), etc.

**Quadrinomial:**An expression which has four terms is called quadrinomial.

Eg. \(x^2-y^2+6+x,\;x^3+3+6x^2+2x\), etc.

**Polynomial:**An expression which has two or more than two terms is called polynomial.

Eg. \(6x^3+x^5+x^4+x^2+x+1\), etc.