IGCSE Maths: Mastering the Laws of Indices, Negative, and Fractional Powers (H)

The Laws of Indices (also known as Powers and Roots) are fundamental rules governing how numbers and algebraic terms interact through multiplication and division. Mastery of these laws is critical for the Edexcel IGCSE 4MA1 Higher Tier syllabus, forming the backbone of algebra, calculus, and topics like Standard Form.

This guide provides a comprehensive overview of Topic 1.5 within the IGCSE Number System, focusing on the rigorous application of the core laws, and the advanced concepts of negative and fractional indices required for Grades 7-9.

Understanding Powers and Roots

Index notation is a shorthand for repeated multiplication.

In the expression aⁿ:

• a is the Base (the number being multiplied).
• n is the Index (or Power/Exponent) (how many times the base is multiplied by itself).

Example: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Roots are the inverse operation of powers. The nth root of a number b (ⁿ√b) is the value a such that aⁿ = b.

• Square Root: √36 = 6.
• Cube Root: ∛8 = 2. (Using ∛ symbol for cube root)

The Fundamental Laws of Indices

The following laws must be memorized and applied fluently. These rules apply when the base is the same.

Advanced Indices: Negative and Fractional (H)

The Higher Tier specification requires a deep understanding of how negative and fractional indices function.

6. Negative Indices (Reciprocals) (H)

Rule: A negative index signifies the reciprocal of the base raised to the positive index. It means "1 divided by."

a^-n = 1aⁿ

Example: 5^-2 = 15² = 125.

Fractions with Negative Indices:
When a fraction is raised to a negative power, flip the fraction (find the reciprocal) and make the power positive.

(ab)^-n = (ba)ⁿ

Example: (23)^-3 = (32)³ = 3³2³ = 278.

7. Fractional Indices (Roots) (H)

Rule: A fractional index represents a root.

Unit Fractions (Numerator = 1): The denominator indicates the type of root.
a¹ⁿ = ⁿ√a

Example: 16¹² = √16 = 4; 27¹³ = ∛27 = 3.

General Fractions (Numerator ≠ 1): The numerator indicates the power, and the denominator indicates the root.
a^mn = (ⁿ√a)^m or ⁿ√(a^m)

Methodology: When evaluating, it is almost always easier to calculate the root first (to make the number smaller), and then apply the power.

Example: Evaluate 8²³.
1. Root (Denominator 3): ∛8 = 2.
2. Power (Numerator 2): 2² = 4.
Result: 4.

Video Lesson: Visualizing Negative Indices & The "Root First" Strategy

Step-by-Step Worked Examples

The following examples demonstrate the application of these laws in IGCSE exam-style questions.

Real-World Application (Global Context)

Indices are fundamental in science, finance, and biology for modeling exponential growth and decay.

Scenario: Radioactive Decay (Half-Life)

The amount of a radioactive substance remaining after a certain time is modeled using indices. The half-life is the time it takes for half of the substance to decay.

Problem: A substance has a half-life of 5 years. If we start with 400g, how much remains after 20 years?

Solution:

The formula for the remaining amount (A) is A = I × (0.5)ⁿ, where I is the initial amount and n is the number of half-lives that have passed.

1. Calculate the number of half-lives (n):
   Total time = 20 years. Half-life = 5 years.
   n = 20 ÷ 5 = 4 half-lives.
2. Apply the formula:
   A = 400 × (0.5)⁴.
   (Using fractions: (0.5) = (12)).
   A = 400 × (12)⁴.
3. Evaluate the index:
   (12)⁴ = 1⁴2⁴ = 116.
4. Calculate the final amount:
   A = 400 × 116 = 25.

Answer: 25g of the substance remains.

Exam Technique and Common Pitfalls

Accuracy with indices requires careful attention to detail and a solid understanding of the rules.

The "Coefficient Trap"

The most common error in algebraic simplification occurs when raising a product to a power.

• Error: (3x²)³ = 3x⁶.
• Fix: Remember the power applies to the coefficient as well. (3x²)³ = 3³ × (x²)³ = 27x⁶.

Confusing the Rules

Students often confuse the Multiplication Law (add indices) and the Power of a Power Law (multiply indices).

• a⁵ × a³ = a⁸ (Add).
• (a⁵)³ = a¹⁵ (Multiply).

Fractional Indices Strategy: Root First!

When evaluating complex fractional indices (e.g., 32³⁵), always calculate the root first to keep the numbers manageable. (⁵√32)³ = 2³ = 8. Calculating the power first (32³) makes the calculation extremely difficult without a calculator.

Negative Bases and BIDMAS

Be extremely careful when dealing with negative bases. The presence or absence of brackets changes the meaning entirely.

• (-3)² = (-3) × (-3) = 9. (The base is -3).
• -3² = -(3 × 3) = -9. (The base is 3; the negative is applied after the power due to BIDMAS).

Developing fluency with these rules is crucial for advanced algebraic manipulation skills required for Grades 8 and 9.

Summary Checklist and Next Steps

Mastery of the Laws of Indices is essential for success in the Higher Tier IGCSE, as these rules underpin many advanced topics.

Checklist:

• [ ] I understand the concepts of Base, Index, Power, and Root.
• [ ] I have memorized and can apply the fundamental laws: Multiplication, Division, and Power of a Power.
• [ ] I know the Zero Index rule (a⁰=1).
• [ ] I can correctly apply powers to products, ensuring coefficients are included.
• [ ] (H) I understand Negative Indices as reciprocals (a^-n = 1aⁿ).
• [ ] (H) I understand Fractional Indices as roots and powers (a^mn = (ⁿ√a)^m).
• [ ] (H) I can evaluate complex numerical expressions involving negative and fractional indices.
• [ ] (H) I can solve equations involving indices by finding a common base (Grade 9 skill).

Next Steps:

The skills learned here are immediately applied in Topic 1.10 (Standard Form) and Topic 1.11 (Surds), which relates closely to fractional indices. They are also fundamental to all subsequent Algebra topics (Section 2).