Informative line

# Expression

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.

## Numerical expression

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

Example: $$2(8+6)$$

### Variable/Algebraic expression

• 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.

#### Which one of the following expressions is an algebraic expression?

A $$2(8+7)$$

B $$7(3\div2)$$

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

D $$3x+2t$$

×

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

In options (A), (B) and (C), there are only numbers and do not contain any variables, so these are numerical expressions.

Option (D) is $$3x+2t$$, which contains variables and numbers connected with operations, so it is an algebraic expression.

Hence, option (D) is correct.

### Which one of the following expressions is an algebraic expression?

A

$$2(8+7)$$

.

B

$$7(3\div2)$$

C

$$3(7)+2(4)$$

D

$$3x+2t$$

Option D is Correct

# Locating the Grouping Symbols (Parentheses)

## Use of grouping symbols (parentheses)

• 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.

#### Which one of the following options represents the correct position of parentheses so that the expression, $$100 \div 5 × 6 - 2$$ gives the result as 80?

A $$(10 0\div 5)× 6 - 2$$

B $$100 \div (5 × 6 )- 2$$

C $$100 \div (5 × 6 - 2)$$

D $$100 \div 5 × (6 - 2)$$

×

Evaluating option (A)

$$(100 \div 5) × 6 -2$$

$$=20 × 6 - 2$$

$$=120 - 2$$

$$= 118$$

Here, the value is not 80.

Thus, option (A) is incorrect.

Evaluating option (B)

$$100 \div (5 × 6 ) - 2$$

$$= 100 \div (30) - 2$$

$$= 3.33- 2$$

$$=1.33$$

Here, the value is not 80.

Thus, option (B) is incorrect.

Evaluating option (C)

$$100 \div (5 × 6 - 2)$$

$$= 100 \div (30- 2)$$

$$= 100 \div 28$$

$$= 3.57$$

Here, the value is not 80.

Thus, option (C) is incorrect.

Evaluating option (D)

$$100 \div 5 × (6-2)$$

$$=100\div 5 × 4$$

$$=20×4$$

$$=80$$

Here, the value is $$80$$.

Thus, option (D) is correct.

### Which one of the following options represents the correct position of parentheses so that the expression, $$100 \div 5 × 6 - 2$$ gives the result as 80?

A

$$(10 0\div 5)× 6 - 2$$

.

B

$$100 \div (5 × 6 )- 2$$

C

$$100 \div (5 × 6 - 2)$$

D

$$100 \div 5 × (6 - 2)$$

Option D is Correct

# PEMDAS Rule

• 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.

"Please Excuse My Dear Aunt Sally"

• 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.

#### Evaluate the expression:  $$35 + 3^2 - (3× 2 ) × 7$$

A $$3$$

B $$2$$

C $$4$$

D $$5$$

×

$$35 + 3^2 - (3 × 2) × 7$$

We shall use the PEMDAS rule to evaluate this expression.

$$35 + 3^2 - (3 × 2) × 7$$

(i) Evaluating parenthesis

$$= 35 +3^2 - 6 × 7$$

(ii) Evaluating exponents

$$= 35 + 9 - 6 × 7$$

(iii) Multiplication

$$= 35 + 9- 42$$

$$= 4 4-42$$

(v) Subtraction

$$= 2$$

Hence, option (B) is correct.

### Evaluate the expression:  $$35 + 3^2 - (3× 2 ) × 7$$

A

$$3$$

.

B

$$2$$

C

$$4$$

D

$$5$$

Option B is Correct

# Types of Algebraic Expressions

• 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.

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.

#### Which one of the following is NOT a polynomial?

A $$2x+3$$

B $$3x$$

C $$3x+5+y$$

D $$1+2x+3y+4z$$

×

A polynomial has two or more than two terms.

Thus, $$3x$$ is not a polynomial.

Hence, option (B) is correct.

### Which one of the following is NOT a polynomial?

A

$$2x+3$$

.

B

$$3x$$

C

$$3x+5+y$$

D

$$1+2x+3y+4z$$

Option B is Correct