Chapter 1.2: Laws of Indices (All Rules)

🎯 Learning Journey

Step 1: Multiplication Law (aᵐ × aⁿ = aᵐ⁺ⁿ)

When multiplying powers with the same base, add the indices. Example: 2³ × 2⁴ = 2⁷

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Step 2: Division Law (aᵐ ÷ aⁿ = aᵐ⁻ⁿ)

When dividing powers with the same base, subtract the indices. Example: 5⁸ ÷ 5³ = 5⁵

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Step 3: Power of Power Law ((aᵐ)ⁿ = aᵐⁿ) & Zero/Negative Powers

Power to a power: multiply indices. Zero power: a⁰ = 1. Negative power: a⁻ⁿ = 1/aⁿ

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Step 4: Apply to GCSE Problem-Solving

Use all laws together to simplify complex expressions and solve algebraic equations.

📖 Understanding the Topic

🎯 What You'll Learn

The laws of indices are fundamental rules that make working with powers efficient and accurate. In Year 9 and GCSE, you'll master ALL the laws including zero and negative indices, which are essential for algebra, standard form, and equations.

📚 The Complete Laws of Indices:

Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ (add the indices)
Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract the indices)
Power of a power: (aᵐ)ⁿ = aᵐⁿ (multiply the indices)
Zero power: a⁰ = 1 (any number to power 0 equals 1)
Negative power: a⁻ⁿ = 1/aⁿ (negative flips to denominator)
Fractional power: a^(1/n) = ⁿ√a (coming in later chapters)

🚀 Why This Matters for GCSE

📝 GCSE Algebra

Index laws appear in nearly every GCSE algebra question - simplifying expressions, factorizing, and solving equations.

🔬 Standard Form

Essential for working with very large and very small numbers in science (10⁶, 10⁻³, etc).

🎯 Higher Tier

Negative and fractional indices are tested extensively in GCSE Higher and are crucial for A-level preparation.

💡 Worked Examples

Example 1: Multiplication Law

Question: Simplify 3⁴ × 3⁵

Solution:

Same base (3), so add the indices:

3⁴ × 3⁵ = 3⁴⁺⁵ = 3⁹

Answer: 3⁹ = 19,683

Example 2: Division Law

Question: Simplify x⁹ ÷ x⁴

Solution:

Same base (x), so subtract the indices:

x⁹ ÷ x⁴ = x⁹⁻⁴ = x⁵

Answer: x⁵

Example 3: Power of a Power

Question: Simplify (2³)⁴

Solution:

Power of a power, multiply the indices:

(2³)⁴ = 2³ˣ⁴ = 2¹²

Answer: 2¹² = 4,096

Example 4: Zero Power (NEW!)

Question: Calculate 7⁰

Solution:

Any number (except 0) to the power 0 equals 1:

7⁰ = 1

This works for ALL bases: 1000⁰ = 1, x⁰ = 1

Answer: 1

Example 5: Negative Power (GCSE Higher)

Question: Calculate 2⁻³

Solution:

Negative power = flip to denominator:

2⁻³ = 1/2³ = 1/8

Answer: 1/8 = 0.125

✏️ Practice Questions

Test your understanding with these GCSE-style questions:

⚠️ Common Mistakes & How to Avoid Them

Learn from typical GCSE errors students make!

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Multiplying Instead of Adding Indices

What students often do wrong:

When seeing 2³ × 2⁴, students calculate 2¹² (multiplying 3×4) instead of 2⁷ (adding 3+4). This is a very common GCSE error.

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How to Avoid This Mistake

Correct approach: When you MULTIPLY powers, you ADD indices. When you raise to a POWER, you MULTIPLY indices.

Memory tip: "Multiply terms = ADD powers. Power of power = MULTIPLY powers."

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Forgetting a⁰ = 1

What students often do wrong:

Students think 5⁰ = 0 or 5⁰ = 5. This loses marks in GCSE exams!

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How to Avoid This Mistake

Correct approach: ANY number (except 0) to the power 0 = 1. Always! 100⁰ = 1, x⁰ = 1, (-5)⁰ = 1

Memory tip: "Zero power = answer is ONE!"

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Negative Powers Confusion

What students often do wrong:

Students calculate 3⁻² = -9 (making the answer negative) instead of 3⁻² = 1/9.

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How to Avoid This Mistake

Correct approach: Negative power means reciprocal (flip to fraction). a⁻ⁿ = 1/aⁿ

Memory tip: "Negative power = positive fraction!"

💡 GCSE Exam Tip

Always check if the bases are the same before applying index laws! 2³ × 3⁴ CANNOT be simplified using index laws because the bases are different (2 and 3).

📋 Chapter Summary

🎉 Congratulations!

You've mastered all the Laws of Indices for GCSE!

🎯 GCSE Skills You've Developed:

✓ Multiply powers with same base (aᵐ × aⁿ = aᵐ⁺ⁿ)

✓ Divide powers with same base (aᵐ ÷ aⁿ = aᵐ⁻ⁿ)

✓ Calculate power of a power ((aᵐ)ⁿ = aᵐⁿ)

✓ Apply zero power rule (a⁰ = 1)

✓ Work with negative indices (a⁻ⁿ = 1/aⁿ) - Higher tier

✓ Simplify complex algebraic expressions using all laws

🚀 What's Next?

Next: Surds - Introduction and simplification (Higher tier)