Indices and Surds

Introduction to Indices

Indices are powers that show how many times a number or algebraic expression is multiplied by itself.

Examples

• 2³ = 2 × 2 × 2 = 8

• 5² = 25

• x⁴ = x × x × x × x

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Basic Laws of Indices

Multiplication Rule

aᵐ × aⁿ = aᵐ⁺ⁿ

Example

x³ × x⁵

= x⁸

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Division Rule

aᵐ/aⁿ = aᵐ⁻ⁿ

Example

x⁷/x²

= x⁵

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Power of a Power Rule

(aᵐ)ⁿ = aᵐⁿ

Example

(x³)²

= x⁶

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Power of a Product Rule

(ab)ⁿ = aⁿbⁿ

Example

(2x)³

= 8x³

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Power of a Fraction Rule

(a/b)ⁿ = aⁿ/bⁿ

Example

(x/2)²

= x²/4

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Zero Index Rule

a⁰ = 1

Where a ≠ 0

Example

5⁰ = 1

x⁰ = 1

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Negative Indices

a⁻ⁿ = 1/aⁿ

Examples

x⁻³ = 1/x³

2⁻² = 1/4

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Converting Negative Indices

Example

3x⁻²

= 3/x²

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Fractional Indices

Square Root Form

a¹⁄² = √a

Example

16¹⁄² = √16 = 4

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Cube Root Form

a¹⁄³ = ∛a

Example

27¹⁄³ = 3

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General Rule

aᵐ⁄ⁿ = ⁿ√(aᵐ)

Example

8²⁄³

= (∛8)²

= 2²

= 4

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Simplifying Index Expressions

Example 1

x³ × x⁴/x²

= x⁷/x²

= x⁵

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Example 2

(2x²y³)²

= 4x⁴y⁶

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Example 3

x⁻²y³/x⁻¹y

= x⁻¹y²

= y²/x

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Scientific Notation

Scientific notation is used to express very large or very small numbers.

Form

a × 10ⁿ

Where:

• 1 ≤ a < 10

• n is an integer

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Examples

• 4500000 = 4.5 × 10⁶

• 0.00032 = 3.2 × 10⁻⁴

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Operations with Scientific Notation

Multiplication

(2 × 10³)(3 × 10⁴)

= 6 × 10⁷

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Division

(8 × 10⁶)/(2 × 10²)

= 4 × 10⁴

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Introduction to Surds

A surd is an irrational root that cannot be simplified into an integer.

Examples

• √2

• √3

• 5√7

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Simplifying Surds

Method

Find perfect square factors.

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Example 1

√12

= √(4 × 3)

= 2√3

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Example 2

√75

= √(25 × 3)

= 5√3

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Adding and Subtracting Surds

Only like surds can be combined.

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Example

3√2 + 5√2

= 8√2

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Example

7√5 − 2√5

= 5√5

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Unlike Surds

2√3 + 4√5

Cannot be simplified further.

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Multiplying Surds

Example 1

√2 × √3

= √6

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Example 2

(2√3)(4√5)

= 8√15

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Expanding Brackets

Example

(√2 + 3)(√2 − 1)

= 2 − √2 + 3√2 − 3

= 2√2 − 1

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Rationalising the Denominator

Remove surds from the denominator.

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Example 1

1/√2

Multiply top and bottom by √2:

= √2/2

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Example 2

3/(2 + √5)

Multiply by conjugate:

= 3(2 − √5)/(2 + √5)(2 − √5)

= (6 − 3√5)/(4 − 5)

= 3√5 − 6

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Conjugates

Conjugates have opposite signs.

Examples

• 2 + √3 and 2 − √3

• 5 − √7 and 5 + √7

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Expanding Surd Expressions

Example

(√5 + 2)(√5 + 4)

= 5 + 4√5 + 2√5 + 8

= 13 + 6√5

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Solving Equations with Indices

Example

2ˣ = 16

Since:

16 = 2⁴

Therefore:

x = 4

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Example

9ˣ = 27

Rewrite with same base:

3²ˣ = 3³

2x = 3

x = 3/2

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Solving Equations with Surds

Example

√(x + 1) = 5

Square both sides:

x + 1 = 25

x = 24

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Surds in Geometry

Pythagoras Example

If sides are:

• 1

• 1

Hypotenuse:

√(1² + 1²)

= √2

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Exact Values

Surds are often used for exact values.

Examples

• sin45° = √2/2

• tan60° = √3

• cos30° = √3/2

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Common Surd Simplifications

Expression	Simplified Form
√8	2√2
√18	3√2
√20	2√5
√27	3√3
√45	3√5
√72	6√2

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Common Index Simplifications

Expression	Simplified Form
x⁴ × x³	x⁷
x⁹/x⁴	x⁵
(x³)²	x⁶
x⁻²	1/x²
x¹⁄²	√x
x³⁄²	x√x

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.

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Common Mistakes in Indices

Mistake	Incorrect	Correct
Adding powers incorrectly	x² + x³ = x⁵	Cannot simplify
Multiplying powers wrongly	x² × x³ = x⁶	x⁵
Negative index error	x⁻² = −x²	1/x²
Fractional index confusion	x¹⁄² = x/2	√x
Zero index error	x⁰ = 0	1

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Common Mistakes in Surds

Mistake	Incorrect	Correct
Adding unlike surds	√2 + √3 = √5	Cannot simplify
Rationalising wrongly	1/√2 = 1√2	√2/2
Squaring surds wrongly	(√3)² = √9	3
Expanding brackets wrongly	(√2 + 1)² = 2 + 1	3 + 2√2

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Exam Tips

• Always leave surds in simplest exact form.

• Rationalise denominators unless instructed otherwise.

• Convert negative indices into positive form.

• Use same bases in index equations whenever possible.

• Check whether surds are like surds before adding.

• Simplify fully before giving final answer.

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Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.