Curriculum

Mathematics — Third Preparatory

Unit 1

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Unit 2

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Unit 3

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Unit 4

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Text lesson

Unit 1: Powers of Powers and Exponential Equations

Mathematics — Third Preparatory

Unit 1: Powers of Powers and Exponential Equations

Explanation

Lesson sections

Overview 1-1 The base, the exponent, and the product and quotient of powers 1-2 The zero exponent and the negative exponent 1-3 Power of a power, of a product and of a quotient 1-4 Sign of a power: even and odd exponents 1-5 Powers, square roots and fractional exponents 1-6 Simplifying expressions that combine several laws 1-7 Exponential equations by equalising the base 1-8 Equations with negative exponents and fractions 1-9 Equations solved by taking out a common factor 1-10 Equations in which the unknown is the base 1-11 Real-life applications 1-12 Summary of the laws of this unit Exam traps and key takeaways

Overview

Use the seven laws of exponents with rational bases, including zero, negative and fractional powers.
Determine the sign of a power from the parity of the exponent.
Solve exponential equations by equalising bases, taking out a common factor or rewriting different bases with a common prime.
Model real-life growth, area and volume problems with exponential equations.

Powers and Exponential Equations
├── 1-1 Laws of powers
│   ├── multiplying and dividing same base
│   ├── zero exponent and negative exponent
│   ├── power of a power, of a product, of a quotient
│   ├── sign: even and odd exponents
│   ├── powers and square roots (fractional exponents)
│   └── simplifying with several laws at once
└── 1-2 Exponential equations
    ├── equalising the base
    ├── negative exponents and fractions
    ├── taking out a common factor
    ├── the unknown is the base
    ├── different bases reduced to a common prime
    ├── real-life applications
    └── equations mixing several ideas

Unit 1: Powers of Powers and Exponential Equations · 1 / 14

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1-1 The base, the exponent, and the product and quotient of powers

Powers of Powers and Exponential: rows of cubes show the laws of exponents and powers: 2³ x 2² = 2⁵, that is 8 x 4 = 32 cubes.

Comparison aspect	Main idea	Keywords
•	In aⁿ a is the base and n the exponent aⁿ means a multiplied by itself n times	baseexponentmeansmultiplied
a^m × aⁿ = a^m+n	same base multiplied: add the exponents	samebasemultipliedexponents
a^m ÷ aⁿ = a^m−n	same base divided: subtract the exponents (a ≠ 0)	samebasedividedsubtract
Write every factor with the same base first	2 × 4ⁿ = 2 × 2²ⁿ = 2²ⁿ⁺¹
•	Note the difference between −aⁿ (the negative of aⁿ) and (−a)ⁿ (the power of the negative base)	Notedifferencebetweennegative

Unit 1: Powers of Powers and Exponential Equations · 2 / 14

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1-2 The zero exponent and the negative exponent

Comparison aspect	Main idea	Keywords
a⁰ = 1	for every a ≠ 0 0⁰ is undefined. So (3x²)⁰ = 1 whenever x ≠ 0	everyundefinedwhenever
a⁻ⁿ = 1/aⁿ	a negative exponent moves the factor to the other side of the fraction bar it does not change the sign of the value	negativeexponentmovesfactor
•	(a/b)⁻ⁿ = (b/a)ⁿ
Example	2⁻³ = 1/8 and (2/5)⁻² = (5/2)² = 25/4

Unit 1: Powers of Powers and Exponential Equations · 3 / 14

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1-3 Power of a power, of a product and of a quotient

(a^m)ⁿ = a^mnmultiply the exponents.

(ab)ⁿ = aⁿ bⁿthe exponent distributes over a product.

(a/b)ⁿ = aⁿ/bⁿlikewise over a quotient (b ≠ 0).

Nested powers((2³)²)⁴ = (2⁶)⁴ = 2²⁴ — always multiply from the inside out.

Unit 1: Powers of Powers and Exponential Equations · 4 / 14

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1-4 Sign of a power: even and odd exponents

(−a)ⁿ is positive when n is even and negative when n is odd.
(−2)⁴ = 16 but −2⁴ = −16; (−2)⁻³ = −1/8.
Any real power of a positive base is positive, so aⁿ = −4 has no real solution when n is even.
For comparison questions, put the numbers over a common exponent or a common base before judging which is larger.

Unit 1: Powers of Powers and Exponential Equations · 5 / 14

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1-5 Powers, square roots and fractional exponents

Comparison aspect	Main idea	Example	Keywords
a1/n) = ⁿ√a	and a^m/n = (ⁿ√a)^m defined for a ≥ 0 when n is even	—	definedeven
•	√(a²) = \|a\| — the absolute value which is why √((−5)²) = 5 and not −5	—	absolutevalue
•	√(ab) = √a · √b for a b ≥ 0 √(a/b) = √a / √b for a ≥ 0 b > 0	—
Simplifying a radical	pull out the largest perfect square	√48 = √(16 × 3) = 4√3	pulllargestperfectsquare
Rationalising a denominator	1/√2 = √2/2 and 1/(√3 + 1) = (√3 − 1)/2	—

Unit 1: Powers of Powers and Exponential Equations · 6 / 14

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1-6 Simplifying expressions that combine several laws

Comparison aspect	Main idea	Keywords
Order of work	powers of powers first then products and quotients of the same base then negative and zero exponents and finally collect like bases	powersfirstthenproducts
Example	(2x³y⁻²)³ / (4x²y)² = (8x⁹y⁻⁶)/(16x⁴y²) = ½ x⁵ y⁻⁸ = x⁵ / (2y⁸)	2x³y4x²y8x⁹y16x⁴y²
•	Leave the answer with positive exponents unless the question asks otherwise	Leaveanswerpositiveexponents

Unit 1: Powers of Powers and Exponential Equations · 7 / 14

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1-7 Exponential equations by equalising the base

Comparison aspect	Main idea	Keywords
If a^x = a^y with a > 0 and a ≠ 1, then x = y	write both sides as powers of the same base and equate the exponents	writebothsidespowers
Example	3^x+1 = 27 → 3^x+1 = 3³ → x = 2
Example	4^x = 8 → 2^2x = 2³ → x = 3/2
•	Check the answer by substitution an exponential equation over a positive base has exactly one real root when the bases match	Checkanswersubstitutionexponential

Unit 1: Powers of Powers and Exponential Equations · 8 / 14

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1-8 Equations with negative exponents and fractions

Clear negative exponents first by taking reciprocals, then equalise bases.
Example 2^−x = 1/16 → 2^−x = 2⁻⁴ → x = 4.
Example (1/3)2x) = 9 → 3−2x) = 3² → x = −1.
Fractional exponents x3/2) = 8 → x = 82/3) = (∛8)² = 4.
To solve an equation with exponents, write both sides with the same base, then set the exponents equal.
For any integer exponent n, use the rule (a/b)n = (b/a)−n, where a and b are non-zero.
For example, (2/3)x+2 = 27/8 can be written as (2/3)x+2 = (3/2)3 = (2/3)−3, so x + 2 = −3 and x = −5.
Another example 9x+2 = 3x. Writing 9 as 32 gives 32x+4 = 3x, so 2x + 4 = x and x = −4. In a third example, 3x+1 = 81 = 34, so x + 1 = 4 and x = 3. Always check the value of x in the original equation.

Unit 1: Powers of Powers and Exponential Equations · 9 / 14

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1-9 Equations solved by taking out a common factor

When powers of the same base are added, factor out the smallest power.
Example 2^x+1 + 2^x = 24 → 2^x (2 + 1) = 24 → 2^x = 8 → x = 3.
Example 5^x+2 − 5^x = 120 → 5^x (25 − 1) = 120 → 5^x = 5 → x = 1.
Never “cancel” a sum of powers by subtracting exponents — that is the trap this method avoids.

Unit 1: Powers of Powers and Exponential Equations · 10 / 14

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1-10 Equations in which the unknown is the base

xⁿ = a → x = a¹/n); for even n there are two real roots ±ⁿ√a when a > 0, and none when a < 0.
Example x² = 49 → x = ±7; x³ = 64 → x = 4; (2x − 1)² = 25 → 2x − 1 = ±5 → x = 3 or x = −2.
Even exponent with a negative right side has no real solution: x⁴ = −16 → ∅.

Break each base into its prime factors 8^x+1 = 16 → 2³⁽^x+1⁾ = 2⁴ → 3x + 3 = 4 → x = 1/3.
If no common prime exists the equation is out of this unit’s scope and needs logarithms.

Unit 1: Powers of Powers and Exponential Equations · 11 / 14

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1-11 Real-life applications

Comparison aspect	Main idea	Keywords
Doubling growth	a colony doubling every hour after n hours is 2ⁿ times the initial count set the ratio equal to a power of 2 and solve for n	colonyeveryhourafter
Area and volume	a square of side 3ᵏ has area 3²ᵏ a cube of side 2ᵏ has volume 2³ᵏ — compare with the laws not with arithmetic multiplication	squaresidecubecompare
Attenuation	a quantity halving each period is (½)ⁿ of its original value	quantityhalvingperiodoriginal
•	Always state the unit and reject a non-integer or negative value of n when the situation is a count of periods	Alwaysstateunitreject

Unit 1: Powers of Powers and Exponential Equations · 12 / 14

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1-12 Summary of the laws of this unit

a^m · aⁿ = a^m+n ; a^m ÷ aⁿ = a^m−n ; (a^m)ⁿ = a^mn ; (ab)ⁿ = aⁿbⁿ ; (a/b)ⁿ = aⁿ/bⁿ.
a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ ; a^m/n = (ⁿ√a)^m.
(−a)ⁿ > 0 for even n, (−a)ⁿ < 0 for odd n.
a^x = a^y ⟹ x = y; xⁿ = a ⟹ x = a¹/n) (two signs for even n).

Unit 1: Powers of Powers and Exponential Equations · 13 / 14

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Exam traps and key takeaways

💡 High-yield traps

−2⁴ = −16 while (−2)⁴ = 16 — brackets decide the sign.
a⁻ⁿ is a reciprocal, not a negative number.
a⁰ = 1 only for a ≠ 0; 0⁰ is undefined.
(a + b)² ≠ a² + b²; the product and quotient laws do not apply to sums.
√(a²) = |a| — forgetting the absolute value loses the sign mark.
Adding powers of the same base means factoring, not adding exponents: 2^x + 2^x = 2 · 2^x = 2^x+1.
Even roots need ± when solving x² = a.
Equalising bases requires a common prime base; matching only the written base (like 4 and 8) is not enough.
In word problems the exponent is a number of periods, so it must be non-negative; reject other values.

Unit 1: Powers of Powers and Exponential Equations · 14 / 14

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