Radical Expressions – Complete Notes, Examples, MCQs and Worksheet

Introduction

A radical expression is an expression that contains a radical symbol (√). Radicals are used to represent roots such as square roots, cube roots, and higher-order roots. Radical expressions are an important part of algebra and are widely used in mathematics, science, and engineering.

Examples:

• √25
• √x
• ³√8
• 2√3
• √(a + b)

What is a Radical?

A radical is a mathematical symbol used to indicate a root.

General form:$$\sqrt[n]{a}$$​

Where:

• n = index (degree of the root)
• a = radicand (number or expression inside the radical sign)

Examples:$$\sqrt{16}=4$$$$\sqrt[3]{27}=3$$ $$\sqrt[4]{81}=3$$

Parts of a Radical Expression

Consider:$$\sqrt[3]{125}$$​

• Radical Symbol: √
• Index: 3
• Radicand: 125

Types of Radical Expressions

1. Square Root

A square root has index 2.$$\sqrt{49}=7$$

2. Cube Root

A cube root has index 3.$$\sqrt[3]{64}=4$$

3. Higher Roots

$$\sqrt[4]{16}=2$$

Simplifying Radical Expressions

A radical expression is simplified when no perfect square (or perfect root) remains inside the radical.

Example 1

Simplify:$$\sqrt{72}$$​

Factor 72:$$72=36\times2$$72=36×2 $$\sqrt{72}=\sqrt{36}\sqrt{2}$$​ $$=6\sqrt{2}$$​

Example 2

Simplify:$$\sqrt{50}$$ $$50=25\times2$$50=25×2 $$\sqrt{50}=5\sqrt{2}$$

Product Property of Radicals

$$\sqrt{ab}=\sqrt a \times \sqrt b$$​

Example:$$\sqrt{12} =\sqrt{4\times3}$$12​=4×3​ $$=2\sqrt3$$

Quotient Property of Radicals

$$\sqrt{\frac{a}{b}} = \frac{\sqrt a}{\sqrt b}$$​​

Example:$$\sqrt{\frac{25}{9}} = \frac{5}{3}$$

Addition and Subtraction of Radical Expressions

Only like radicals can be added or subtracted.

Example

$$3\sqrt2+5\sqrt2$$​ $$=8\sqrt2$$​

Example

$$7\sqrt5-2\sqrt5$$​ $$=5\sqrt5$$

Not Possible

$$\sqrt2+\sqrt3$$​

cannot be simplified further.

Multiplication of Radical Expressions

Example

$$(2\sqrt3)(4\sqrt5)$$ $$=8\sqrt{15}$$

Division of Radical Expressions

Example

$$\frac{12\sqrt6}{3\sqrt2}$$​​ $$=4\sqrt3$$​

Rationalizing the Denominator

A denominator should not contain a radical.

Example

$$\frac{1}{\sqrt2}$$​

Multiply numerator and denominator by √2:$$\frac{1}{\sqrt2}\times\frac{\sqrt2}{\sqrt2}$$​​ $$=\frac{\sqrt2}{2}$$

Applications of Radical Expressions

Radical expressions are used in:

1. Geometry
2. Algebra
3. Physics
4. Engineering
5. Construction
6. Computer Graphics

Solved Examples

Example 1

Simplify:$$\sqrt{98}$$​ $$= \sqrt{49\times2}$$​ $$=7\sqrt2$$​

Example 2

Simplify:$$\sqrt{75}$$​ $$=\sqrt{25\times3}$$​ $$=5\sqrt3$$​

Example 3

Add:$$2\sqrt7+6\sqrt7$$​ $$=8\sqrt7$$​

Example 4

Multiply:$$3\sqrt2 \times 2\sqrt5$$​ $$=6\sqrt10$$

Short Notes on Radical Expressions

• A radical expression contains a radical symbol (√).
• The number inside the radical is called the radicand.
• The small number on the radical is called the index.
• Square roots have index 2.
• Cube roots have index 3.
• Like radicals can be added and subtracted.
• Radicals can be simplified by factoring out perfect squares.
• Rationalizing the denominator removes radicals from denominators.
• Radical expressions are commonly used in algebra and geometry.

MCQs on Radical Expressions

1. Which symbol represents a radical?

A) +
B) √
C) ×
D) ÷

Answer: B

2. What is √81?

A) 8
B) 9
C) 7
D) 6

Answer: B

3. Simplify √64.

A) 6
B) 7
C) 8
D) 9

Answer: C

4. Simplify √50.

A) 5√2
B) 2√5
C) 10√2
D) 25

Answer: A

5. What is ³√27?

A) 2
B) 3
C) 4
D) 9

Answer: B

6. Simplify 4√3 + 2√3.

A) 6√3
B) 8√3
C) 2√3
D) √3

Answer: A

7. Simplify √100.

A) 20
B) 5
C) 10
D) 50

Answer: C

8. Which is a radical expression?

A) x + 5
B) 3x²
C) √x
D) x − 1

Answer: C

9. Simplify √49.

A) 6
B) 7
C) 8
D) 9

Answer: B

10. Simplify 2√5 × 3√2.

A) 5√10
B) 6√10
C) 10√6
D) 12√2

Answer: B

Worksheet / Assignment

Part A: Simplify

1. √36
2. √64
3. √81
4. √45
5. √72
6. √98
7. √108
8. ³√64
9. ³√125
10. ⁴√16

Part B: Add or Subtract

1. 2√3 + 5√3
2. 8√2 − 3√2
3. 6√7 + 4√7
4. 10√5 − 2√5
5. 9√11 + √11

Part C: Multiply

1. (2√3)(4√2)
2. (3√5)(2√7)
3. (5√2)(6√3)
4. (4√6)(2√5)
5. (7√2)(3√8)

Part D: Rationalize

1. 1/√2
2. 3/√5
3. 5/√7
4. 2/√3
5. 4/√11

Part E: Word Problems

1. Find the side length of a square whose area is 144 cm².
2. A square garden has an area of 225 m². Find its side length.
3. Calculate the square root of 400 and explain its meaning in geometry.

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