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Indices

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Week Six - Indices (Exponents)

📋 Table of Contents

Laws of Indices
Product of Indices (Power of a Power)
Fractional Indices
Solving Equations Involving Indices
General Evaluation
Key Points to Remember

📖 Introduction

In mathematics, indices (singular: index), also known as exponents or powers, provide a shorthand way of expressing repeated multiplication of a number by itself.

For example:

2 \times 2 \times 2 \times 2 \times 2 = 2 5 = 32

Here, 2 is called the base and 5 is called the index (or exponent). The expression 2⁵ is read as "2 raised to the power of 5."

General Notation:
xⁿ = x × x × x × ··· × x (n times)
where x = base, n = index

📐 SECTION 1: LAWS OF INDICES

The laws of indices are rules that govern how expressions involving powers can be simplified. These laws hold for all values of the base x ≠ 0, and for all real values of the indices a and b.

1 First Law — Multiplication Law (Same Base)

x^a × x^b = x^a+b

When multiplying two powers that have the same base, add the indices.

Illustration:
x³ × x² = (x × x × x) × (x × x) = x³⁺² = x⁵

2 Second Law — Division Law (Same Base)

x^a ÷ x^b = x^ax^b = x^a−b

When dividing two powers that have the same base, subtract the index of the denominator from the index of the numerator.

Illustration:
x⁵x² = x × x × x × x × xx × x = x⁵⁻² = x³

3 Third Law — Zero Index

x⁰ = 1 (x ≠ 0)

Any non-zero number raised to the power of zero equals 1.

Proof using Second Law:

x a \div x a = x a-a = x 0

But we also know that any number divided by itself equals 1:

x a x a = 1

Therefore: x⁰ = 1

4 Fourth Law — Negative Index

x^−a = 1x^a

A negative index indicates the reciprocal of the base raised to the corresponding positive index.

Proof using Second Law:

x -a = x 0-a = x 0 x a = 1 x a

Example: 2⁻³ = 12³ = 18

✏️ Worked Examples — Laws 1 to 4

Simplify each of the following:

10⁵ × 10⁴
a³ × a⁴
m⁸ ÷ m⁵
24x⁶ ÷ 8x⁴
19⁸ ÷ 19⁸

Solution 1: 10⁵ × 10⁴

Applying the first law (add the indices):

105 \times 10 4 = 10 5+4 = 109

Solution 2: a³ × a⁴

a 3 \times a 4 = a 3+4 = a 7

Solution 3: m⁸ ÷ m⁵

Applying the second law (subtract the indices):

m 8 \div m 5 = m 8-5 = m 3

Solution 4: 24x⁶ ÷ 8x⁴

Divide the coefficients and apply the second law to the variable:

24x 6 8x 4 = 248 \times x 6-4 = 3 \times x 2 = 3x 2

Solution 5: 19⁸ ÷ 19⁸

198 \div 19 8 = 19 8-8 = 19 0 = 1

📝 Evaluation — Practice Questions (Laws 1 to 4)

Simplify each of the following:

6 × z⁰
4⁻³
z³ × (⅙)¹
r × r × r × r⁻⁵

Click to reveal answers

(a) 6 × z⁰ = 6 × 1 = 6

(b) 4⁻³ = 14³ = 164 = 1/64

(c) z³ × 16 = z³6

(d) r × r × r × r⁻⁵ = r^1+1+1+(−5) = r³⁻⁵ = r⁻² = 1r²

🔢 SECTION 2: PRODUCT OF INDICES (Power of a Power)

5 Fifth Law — Power of a Power

(x^a)^b = x^a×b = x^ab

When a power is raised to another power, multiply the indices.

Illustration:
(x²)³ = x² × x² × x² = x²⁺²⁺² = x^2×3 = x⁶

6 Sixth Law — Power of a Product

(xy)^a = x^a · y^a

When a product of bases is raised to a power, the power applies to each factor separately.

Illustration:
(2x)³ = 2³ × x³ = 8x³

7 Seventh Law — Power of a Quotient

(xy)^a = x^ay^a (y ≠ 0)

When a fraction is raised to a power, the power applies to both the numerator and the denominator.

Illustration:
(34)² = 3²4² = 916

📊 Summary Table of All Laws of Indices

Law	Rule	Name
1st	x^a × x^b = x^a+b	Multiplication Law
2nd	x^a ÷ x^b = x^a−b	Division Law
3rd	x⁰ = 1	Zero Index Law
4th	x^−a = 1/x^a	Negative Index Law
5th	(x^a)^b = x^ab	Power of a Power
6th	(xy)^a = x^ay^a	Power of a Product
7th	(x/y)^a = x^a/y^a	Power of a Quotient

✏️ Worked Examples — Power of a Power

Simplify each of the following:

(x²)³
(y⁴)²
(3⁻²)⁻³
(−3d³)²
a⁶ · (−a)⁻⁴

Solution 1: (x²)³

Multiply the indices:

(x 2) 3 = x 2\times3 = x 6

Solution 2: (y⁴)²

(y 4) 2 = y 4\times2 = y 8

Solution 3: (3⁻²)⁻³

3 (-2)\times(-3) = 3 +6 = 3 6

Now compute 3⁶:

36 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 9 \times 9 \times 9 = 27 \times 27 = 729

Solution 4: (−3d³)²

Apply the sixth law — distribute the power to each factor:

(-3d 3) 2 = (-3) 2 \times (d 3) 2

= (-3 \times -3) \times d 3\times2 = 9 \times d 6 = 9d 6

📌 Note: (−3)² = +9 because the product of two negative numbers is positive.

Solution 5: a⁶ · (−a)⁻⁴

First, apply the negative index law:

a 6 \times (-a) -4 = a 6 \times 1 (-a) 4

Now evaluate (−a)⁴:

(-a) 4 = (-a) \times (-a) \times (-a) \times (-a) = a 4

(Even power of a negative base gives a positive result.)

Therefore:

a 6 a 4 = a 6-4 = a 2

📝 Evaluation — Practice Questions (Power of a Power)

Simplify each of the following:

(h⁴)⁻⁵
(−4u²v)³
(−x³)² ÷ x⁴
−(d²) ÷ d⁴ × (−d)
(−c)² × c⁴ ÷ (−c³)

Click to reveal answers

1. (h⁴)⁻⁵ = h^4×(−5) = h⁻²⁰ = 1h²⁰

2. (−4u²v)³ = (−4)³ × (u²)³ × v³ = −64 × u⁶ × v³ = −64u⁶v³

3. (−x³)² ÷ x⁴ = (−1)² · x⁶ ÷ x⁴ = x⁶ ÷ x⁴ = x⁶⁻⁴ = x²

4. −d² ÷ d⁴ × (−d) = −d²⁻⁴ × (−d) = −d⁻² × (−d) = d⁻²⁺¹ = d⁻¹ = 1d

5. (−c)² × c⁴ ÷ (−c³) = c² × c⁴ ÷ (−c³) = c⁶−c³ = −c⁶⁻³ = −c³

🔢 SECTION 3: FRACTIONAL INDICES

Fractional indices involve exponents that are fractions. They are closely related to roots (square roots, cube roots, etc.).

Understanding x^1/n — The nth Root

Square Root

The square root of x, written √x, is the number which when multiplied by itself gives x:

\sqrtx \times \sqrtx = x

Derivation:

Let √x = x^p. Then:

x p \times x p = x 1 x 2p = x 1 Equating the indices: 2p = 1 ⟹ p = ½

√x = x^1/2

Illustration: √25 = 25^1/2 = 5 since 5 × 5 = 25

Cube Root

The cube root of x, written ∛x, is the number which when multiplied by itself three times gives x:

∛x \times ∛x \times ∛x = x

Derivation:

Let ∛x = x^q. Then:

x q \times x q \times x q = x 1 x 3q = x 1 Equating the indices: 3q = 1 ⟹ q = ⅓

∛x = x^1/3

Illustrations:
• ∛8 = 8^1/3 = 2 since 2 × 2 × 2 = 8
• ∛(−27) = (−27)^1/3 = −3 since (−3) × (−3) × (−3) = −27

General nth Root

x^1/n = ⁿ√x
The fractional index 1/n means "take the nth root" of the base.

Expression	Meaning	Value
16^1/2	√16	4
27^1/3	∛27	3
81^1/4	⁴√81	3
32^1/5	⁵√32	2

Understanding x^a/b — General Fractional Index

x^a/b = ^b√(x^a) = (^b√x)^a

A fractional index a/b can be interpreted in two equivalent ways:

Root first, then power: Take the bth root of x, then raise the result to the power a.
Power first, then root: Raise x to the power a, then take the bth root.

Illustration:

8^2/3 = (∛8)² = (2)² = 4

OR equivalently:

8^2/3 = ∛(8²) = ∛64 = 4

💡 Tip: Taking the root first is usually easier because it keeps the numbers smaller!

✏️ Worked Examples — Fractional Indices

Simplify each of the following:

8^−2/3
4^1/6 × 4^1/3
(16/81)^−3/4
√(72a³b⁻² / 2a⁵b⁻⁶)

Solution 1: 8^−2/3

Apply the negative index law first:

8 -2/3 = 1 8 2/3

Now evaluate 8^2/3:

8 2/3 = (∛8) 2 = (2) 2 = 4

Therefore:

8 -2/3 = 14

Solution 2: 4^1/6 × 4^1/3

Apply the first law (add the indices since the bases are the same):

4 1/6 \times 4 1/3 = 4 1/6 + 1/3

Find a common denominator for the fractions:

16 + 13 = 16 + 26 = 36 = 12

Therefore:

4 1/2 = \sqrt4 = 2

Solution 3: (16/81)^−3/4

Apply the negative index law (flip the fraction):

(1681) -3/4 = (8116) 3/4

Now evaluate:

(8116) 3/4 = (4 \sqrt 8116) 3

Find the 4th roots:

4 \sqrt81 = 3 (since 3 4 = 81) 4 \sqrt16 = 2 (since 2 4 = 16)

Therefore:

(32) 3 = 3323 = 278

Solution 4: √(72a³b⁻² / 2a⁵b⁻⁶)

Step 1: Simplify the expression inside the square root.

Divide the coefficients: 72 ÷ 2 = 36

Apply the second law to each variable:

a 3-5 = a -2 b -2-(-6) = b -2+6 = b 4

So the expression inside the square root becomes:

\sqrt(36a -2 b 4)

Step 2: Apply the square root (raise everything to the power ½):

(36a -2 b 4) 1/2 = 36 1/2 \times a -2\times½ \times b 4\times½

= 6 \times a -1 \times b 2

= 6b 2 a

📝 Evaluation — Practice Questions (Fractional Indices)

Simplify each of the following:

(125)^−1/3
(18/32)^−3/2
(∛4)^1.5
64^−5/6
√(1 ⁹/₁₆)

Click to reveal answers

1. 125^−1/3 = 1125^1/3 = 1∛125 = 15 = 1/5 (since 5 × 5 × 5 = 125)

2. (18/32)^−3/2
First simplify: 18/32 = 9/16
(9/16)^−3/2 = (16/9)^3/2 = (√(16/9))³ = (4/3)³ = 64/27

3. (∛4)^1.5 = (4^1/3)^3/2 = 4^(1/3)×(3/2) = 4^1/2 = √4 = 2

4. 64^−5/6 = 164^5/6 = 1(⁶√64)⁵ = 12⁵ = 1/32 (since 2⁶ = 64)

5. √(1 ⁹/₁₆) = √(25/16) = √25√16 = 5/4

🧮 SECTION 4: SOLVING EQUATIONS INVOLVING INDICES

To solve equations involving indices, we use the following key strategies:

🎯 Key Strategies:

Strategy 1: If x^a = x^b (same base), then a = b (equate the indices).
Strategy 2: Express both sides of the equation with the same base, then equate the indices.
Strategy 3: Use the laws of indices to simplify, then solve.
Strategy 4: Raise both sides to an appropriate power to eliminate fractional indices.

✏️ Worked Examples — Solving Equations

Solve the following equations:

2r⁻³ = −16
5x ÷ 5 = 40x^−1/2 ÷ 5
4^c−1 = 64

Solution 1: 2r⁻³ = −16

Step 1: Divide both sides by 2:

r -3 = -16 2 = -8

Step 2: Apply the negative index law:

r -3 = -8 ⟹ 1 r 3 = -8

Step 3: Cross multiply:

1 = -8r 3 r 3 = 1 -8 = - 18

Step 4: Take the cube root of both sides:

r = ∛(- 18) = ∛(-1) ∛8 = -1 2

r = -½

✅ Verification:
2(−½)⁻³ = 2 × 1(−½)³ = 2 × 1−⅛ = 2 × (−8) = −16 ✓

Solution 2: 5x ÷ 5 = 40x^−1/2 ÷ 5

Step 1: Simplify both sides:

x = 8x -1/2

Step 2: Apply the negative index law:

x = 8 x 1/2

Step 3: Cross multiply:

x \times x 1/2 = 8

Step 4: Apply the first law (add indices):

x 1 + ½ = 8 x 3/2 = 8

Step 5: Raise both sides to the power 2/3:

(x 3/2) 2/3 = 8 2/3 x (3/2)\times(2/3) = 8 2/3 x 1 = (∛8) 2 = (2) 2 = 4

x = 4

✅ Verification:
LHS = 5(4) ÷ 5 = 4
RHS = 40(4)^−1/2 ÷ 5 = 40 × ½ ÷ 5 = 20 ÷ 5 = 4 ✓

Solution 3: 4^c−1 = 64

Step 1: Express both sides with the same base.
Since 4 = 4 and 64 = 4³:

4 c-1 = 4 3

Step 2: Since the bases are equal, equate the indices:

c - 1 = 3

Step 3: Solve for c:

c = 3 + 1

c = 4

✅ Verification: 4⁴⁻¹ = 4³ = 64 ✓

📝 Evaluation — Practice Questions (Solving Equations)

Solve the following equations:

a^2/3 = 9
2x³ = 54

Click to reveal answers

1. a^2/3 = 9
Raise both sides to the power 3/2:
a = 9^3/2 = (√9)³ = 3³ = 27

2. 2x³ = 54
Divide both sides by 2: x³ = 27
Take the cube root: x = ∛27 = 3

🏆 GENERAL EVALUATION

Solve or simplify the following:

Question 1

If 9^2x+13^x = 81^x−2, find x.

Solution:

Express everything in base 3. Recall: 9 = 3² and 81 = 3⁴.

(3 2) 2x+1 3 x = (3 4) x-2

3 2(2x+1) 3 x = 3 4(x-2)

3 4x+2 3 x = 3 4x-8

Apply the second law on the left:

3 (4x+2)-x = 3 4x-8

3 3x+2 = 3 4x-8

Equate the indices:

3x + 2 = 4x - 8 2 + 8 = 4x - 3x x = 10

Question 2

Solve: 9^x−1 = 27^x+1

Solution:

Express both sides in base 3. Recall: 9 = 3² and 27 = 3³.

(3 2) x-1 = (3 3) x+1

3 2(x-1) = 3 3(x+1)

3 2x-2 = 3 3x+3

Equate the indices:

2x - 2 = 3x + 3 -2 - 3 = 3x - 2x x = -5

Question 3

Simplify: ∛(72p⁻³q⁻⁷9p⁹q⁵)

Solution:

Step 1: Simplify inside the cube root.

Divide the coefficients: 72 ÷ 9 = 8

Apply the second law to each variable:

p -3-9 = p -12 q -7-5 = q -12

So the expression becomes:

∛(8p -12 q -12)

Step 2: Apply the cube root (raise to power 1/3):

(8p -12 q -12) 1/3 = 8 1/3 \times p -12\times⅓ \times q -12\times⅓

= 2 \times p -4 \times q -4

= 2 p 4 q 4

2 p 4 q 4

🔑 KEY POINTS TO REMEMBER

Concept	Formula / Rule
Multiplication (same base)	x^a × x^b = x^a+b
Division (same base)	x^a ÷ x^b = x^a−b
Zero index	x⁰ = 1
Negative index	x^−a = 1/x^a
Power of a power	(x^a)^b = x^ab
Power of a product	(xy)^a = x^ay^a
Fractional index (root)	x^1/n = ⁿ√x
General fractional index	x^a/b = (^b√x)^a
Solving: same base	If x^a = x^b, then a = b

💡 Remember: The key to mastering indices is practice. Always identify the base, apply the correct law, and simplify step by step. When solving equations, try to express both sides with the same base before equating the indices.

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Indices

INDICES (EXPONENTS)

📋 Table of Contents

📖 Introduction

✏️ Worked Examples — Laws 1 to 4

📝 Evaluation — Practice Questions (Laws 1 to 4)

📊 Summary Table of All Laws of Indices

✏️ Worked Examples — Power of a Power

📝 Evaluation — Practice Questions (Power of a Power)

Understanding x1/n — The nth Root

Square Root

Cube Root

General nth Root

Understanding xa/b — General Fractional Index

✏️ Worked Examples — Fractional Indices

📝 Evaluation — Practice Questions (Fractional Indices)

🎯 Key Strategies:

✏️ Worked Examples — Solving Equations

📝 Evaluation — Practice Questions (Solving Equations)

Question 1

Question 2

Question 3

Test yourself on Mathematics

Quadratic Equations II

Quadratic Equations

SIMPLE EQUATION AND VARIATIONS

LOGARITHMS II

Logarithms I

Indices

Understanding x^1/n — The nth Root

Understanding x^a/b — General Fractional Index