📐Math 2: Laws of Integral Indices

Pre-DSE Mathematics — A Complete Study Guide (DSE M1 Module)

2.1 What is an Index?

📐 Definition

An index (also called an exponent or power) indicates how many times a number is multiplied by itself.

aⁿ = a × a × a × … × a (n copies of a)

Where:

a = the base
n = the index, exponent, or power
The whole expression aⁿ is called a power of a

Examples:

2³ = 2 × 2 × 2 = 8 (2 multiplied by itself 3 times)
5⁴ = 5 × 5 × 5 × 5 = 625
x⁵ = x · x · x · x · x

📌 Key Point: This chapter covers only integral indices — positive integers, zero, and negative integers. Fractional indices (such as x^1/2) are introduced later in DSE M2.

2.2 The Five Laws of Indices

This is the core of DSE Mathematics Module 1. You must memorise every one of them — they will appear on the exam every year.

📐 Law 1: Multiplication Law (same base)

a^m × aⁿ = a^{m + n}

Question: Simplify 2³ × 2⁵

① The bases are the same (both 2), so add the indices.

② 3 + 5 = 8

③ Answer: 2⁸ = 256

📐 Law 2: Division Law (same base)

a^m ÷ aⁿ = a^{m − n}

Question: Simplify 5⁷ ÷ 5⁴

① The bases are the same (both 5), so subtract the indices.

② 7 − 4 = 3

③ Answer: 5³ = 125

📐 Law 3: Power of a Power

(a^m)ⁿ = a^mn

Question: Simplify (x²)³

① Multiply the outer index into the inner index.

② 2 × 3 = 6

③ Answer: x⁶

📐 Law 4: Power of a Product

(ab)ⁿ = aⁿ × bⁿ

Question: Simplify (2xy)³

① Raise each factor to the power separately.

② 2³ × x³ × y³

③ Answer: 8x³y³

📐 Law 5: Power of a Quotient

(a / b)ⁿ = aⁿ / bⁿ (b ≠ 0)

Question: Simplify (x / y)³

① Raise numerator and denominator to the power separately.

② Answer: x³ / y³

💡 Memory Mnemonic: "Multiply → Add, Divide → Subtract, Power → Multiply, Product → Distribute."

2.3 Zero Index and Negative Index

📐 Zero Index

a⁰ = 1 (any non-zero number raised to the power zero equals 1)

Examples:

5⁰ = 1
100⁰ = 1
(2x)⁰ = 1 (provided 2x ≠ 0)

⚠️ Note: 0⁰ is undefined! Do not write 1.

📐 Negative Index

a⁻ⁿ = 1 / aⁿ (a negative index gives the reciprocal)

Question: Simplify 2⁻³

① Apply the formula a⁻ⁿ = 1 / aⁿ.

② Answer: 1 / 2³ = 1/8

Question: Simplify x⁻⁵

① Answer: 1 / x⁵

Question: Simplify 1 / 2⁻³ (a DSE favourite!)

① Reciprocal of the reciprocal: 1 / 2⁻³ = 2³.

② Answer: 8

📌 Important Contrast:

2⁻³ = 1/8 (less than 1)
1 / 2⁻³ = 8 (greater than 1)
Many students confuse these two — treat the negative sign in the numerator and the index as separate.

2.4 Comprehensive Simplification (DSE Standard)

📌 Typical DSE Question Type

The most common DSE M1 question is: "You are given a long expression. Simplify it to its most compact form." The standard approach is to handle brackets first, then combine like bases, then rewrite into positive-index form.

Question 1: Simplify (2x³y⁻²)² ÷ (4x⁻¹y⁻³)

① Remove the bracket first: (2x³y⁻²)² = 4x⁶y⁻⁴

② The expression becomes: 4x⁶y⁻⁴ ÷ (4x⁻¹y⁻³)

③ Divide the coefficients: 4 / 4 = 1

④ Index of x: 6 − (−1) = 7 → x⁷

⑤ Index of y: −4 − (−3) = −1 → y⁻¹ = 1/y

⑥ Answer: x⁷ / y

Question 2: Simplify (a²b⁻³c)⁴ × (a⁻¹b²c⁻²)⁻²

① First bracket: (a²b⁻³c)⁴ = a⁸b⁻¹²c⁴

② Second bracket: (a⁻¹b²c⁻²)⁻² = a²b⁻⁴c⁴

③ Multiply: a^{8 + 2} × b^{−12 − 4} × c^{4 + 4} = a¹⁰ × b⁻¹⁶ × c⁸

④ Answer: a¹⁰ c⁸ / b¹⁶

Question 3: Simplify (x²y)³ / (x⁻¹y²)²

① Numerator: (x²y)³ = x⁶y³

② Denominator: (x⁻¹y²)² = x⁻²y⁴

③ Divide: x^{6 − (−2)} × y^{3 − 4} = x⁸ × y⁻¹

④ Answer: x⁸ / y

Question 4: Simplify ((a⁻²)³ × b⁴) / (a²b⁻¹)⁻¹

① Numerator: (a⁻²)³ × b⁴ = a⁻⁶ × b⁴

② Denominator: (a²b⁻¹)⁻¹ = a⁻² × b¹

③ Divide: a^{−6 − (−2)} × b^{4 − 1} = a⁻⁴ × b³

④ Answer: b³ / a⁴

📌 The Three-Step DSE Simplification Procedure:

Remove all brackets first — apply the Power-of-a-Power and Power-of-a-Product laws.
Combine like bases — use the Multiplication and Division laws.
Standardise the final form — rewrite negative indices as reciprocals, keep positive indices in the numerator.

2.5 Common Pitfalls

⚠️ Pitfall 1: Forgetting the Sign Change

A negative index a⁻ⁿ means "reciprocal," not "negative."

❌ 2⁻³ = −8 (Wrong!)

✅ 2⁻³ = 1/8 (Correct)

⚠️ Pitfall 2: Confusing Coefficients and Indices

The numerical coefficient (the big number in front) and the index (the small superscript) must be treated separately.

❌ (2x)³ = 2x³ (Forgot to raise the 2 as well!)

✅ (2x)³ = 2³ × x³ = 8x³

⚠️ Pitfall 3: Dropping a Letter When Distributing

When expanding a power, every letter must be raised to the power.

(a²b)² = a⁴b² (both a and b get raised)

⚠️ Pitfall 4: The Mystery of 0⁰

0⁰ is undefined in mathematics. Do not write 1.

⚠️ Pitfall 5: Wrong Order in Division

For division, the index is "dividend index − divisor index," not the other way around.

❌ a³ ÷ a⁵ = a^{5 − 3} = a² (Wrong)

✅ a³ ÷ a⁵ = a^{3 − 5} = a⁻² = 1 / a² (Correct)

🎯 The Six Formulas You Must Memorise

Multiplication: a^m × aⁿ = a^{m + n}
Division: a^m ÷ aⁿ = a^{m − n}
Power of a Power: (a^m)ⁿ = a^mn
Power of a Product: (ab)ⁿ = aⁿ × bⁿ
Zero Index: a⁰ = 1 (a ≠ 0)
Negative Index: a⁻ⁿ = 1 / aⁿ

📊 Quick-Reference: Five Question Types

#	Question Type	Recognising Feature	Key Strategy
1	Zero index	a⁰	Answer = 1
2	Negative index	a⁻ⁿ	Answer = 1 / aⁿ
3	Multiplication (same base)	a^m × aⁿ	Add the indices
4	Division (same base)	a^m ÷ aⁿ	Subtract the indices
5	Comprehensive simplification	Brackets + multiple letters	Remove brackets → Combine → Standardise

📝 Key Vocabulary (Chinese ↔ English)

Chinese	English	Symbol
指數	Index / Exponent / Power	n
底數	Base	a
冪	Power	aⁿ
指數定律	Laws of Indices	—
化簡	Simplify	—

📝 Quick Practice — Test Yourself!

Question 1: Simplify 3⁴ × 3²

📌 Click to reveal answer

Same base multiplied → add the indices: 4 + 2 = 6

Answer: 3⁶ = 729

Question 2: Simplify 7⁵ ÷ 7²

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Same base divided → subtract the indices: 5 − 2 = 3

Answer: 7³ = 343

Question 3: Simplify (5²)³

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Power of a power → multiply the indices: 2 × 3 = 6

Answer: 5⁶ = 15,625

Question 4: Simplify 2⁻⁴

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Negative index → reciprocal

Answer: 1 / 2⁴ = 1/16

Question 5: Simplify 1 / 5⁻²

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Reciprocal of a reciprocal returns the original

Answer: 5² = 25

Question 6: Simplify (2x)⁴

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Power of a product — raise the coefficient as well

Answer: 2⁴ × x⁴ = 16x⁴

Question 7: Simplify (x²y³)²

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Index of x: 2 × 2 = 4, index of y: 3 × 2 = 6

Answer: x⁴y⁶

Question 8: Simplify a³ × a⁻⁵

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Add the indices: 3 + (−5) = −2

Answer: a⁻² = 1 / a²

Question 9: Simplify (3x²y⁻¹)² ÷ (x⁻¹y)³

📌 Click to reveal answer

Numerator: (3x²y⁻¹)² = 9x⁴y⁻²

Denominator: (x⁻¹y)³ = x⁻³y³

Divide: coefficient 9/1 = 9, x index 4 − (−3) = 7, y index −2 − 3 = −5

Answer: 9x⁷ / y⁵

Question 10: Simplify ((a²b⁻¹)⁻² × (a⁻¹b)³) / (a³b⁻²)

📌 Click to reveal answer

First numerator bracket: (a²b⁻¹)⁻² = a⁻⁴b²

Second numerator bracket: (a⁻¹b)³ = a⁻³b³

Numerator combined: a^{−4 − 3} × b^{2 + 3} = a⁻⁷b⁵

Denominator: a³b⁻²

Divide: a^{−7 − 3} × b^{5 − (−2)} = a⁻¹⁰ × b⁷

Answer: b⁷ / a¹⁰