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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

Overview

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

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.
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 aspectMain ideaKeywords
•
  • In an
  • a is the base and n the exponent
  • an means a multiplied by itself n times
baseexponentmeansmultiplied
am × an = am+nsame base multiplied: add the exponentssamebasemultipliedexponents
am ÷ an = am−nsame base divided: subtract the exponents (a ≠ 0)samebasedividedsubtract
Write every factor with the same base first2 × 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
1-2 The zero exponent and the negative exponent

1-2 The zero exponent and the negative exponent

Comparison aspectMain ideaKeywords
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
1-3 Power of a power, of a product and of a quotient

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

(am)n = amnmultiply 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
1-4 Sign of a power: even and odd exponents

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
1-5 Powers, square roots and fractional exponents

1-5 Powers, square roots and fractional exponents

Comparison aspectMain ideaExampleKeywords
a1/n) = n√a
  • and am/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 radicalpull out the largest perfect square√48 = √(16 × 3) = 4√3pulllargestperfectsquare
Rationalising a denominator1/√2 = √2/2 and 1/(√3 + 1) = (√3 − 1)/2—
Unit 1: Powers of Powers and Exponential Equations · 6 / 14
1-6 Simplifying expressions that combine several laws

1-6 Simplifying expressions that combine several laws

Comparison aspectMain ideaKeywords
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 otherwiseLeaveanswerpositiveexponents
Unit 1: Powers of Powers and Exponential Equations · 7 / 14
1-7 Exponential equations by equalising the base

1-7 Exponential equations by equalising the base

Comparison aspectMain ideaKeywords
If ax = ay with a > 0 and a ≠ 1, then x = ywrite both sides as powers of the same base and equate the exponentswritebothsidespowers
Example3x+1 = 27 → 3x+1 = 33 → x = 2
Example4x = 8 → 22x = 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
1-8 Equations with negative exponents and fractions

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
1-9 Equations solved by taking out a common factor

1-9 Equations solved by taking out a common factor

  • When powers of the same base are added, factor out the smallest power.
  • Example 2x+1 + 2x = 24 → 2x (2 + 1) = 24 → 2x = 8 → x = 3.
  • Example 5x+2 − 5x = 120 → 5x (25 − 1) = 120 → 5x = 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
1-10 Equations in which the unknown is the base

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 8x+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
1-11 Real-life applications

1-11 Real-life applications

Comparison aspectMain ideaKeywords
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
Attenuationa quantity halving each period is (½)n of its original valuequantityhalvingperiodoriginal
•Always state the unit and reject a non-integer or negative value of n when the situation is a count of periodsAlwaysstateunitreject
Unit 1: Powers of Powers and Exponential Equations · 12 / 14
1-12 Summary of the laws of this unit

1-12 Summary of the laws of this unit

  • am · an = am+n ; am ÷ an = am−n ; (am)n = amn ; (ab)n = anbn ; (a/b)n = an/bn.
  • a0 = 1 (a ≠ 0); a−n = 1/an ; am/n = (n√a)m.
  • (−a)n > 0 for even n, (−a)n < 0 for odd n.
  • ax = ay ⟹ x = y; xn = a ⟹ x = a1/n) (two signs for even n).
Unit 1: Powers of Powers and Exponential Equations · 13 / 14
Exam traps and key takeaways

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: 2x + 2x = 2 · 2x = 2x+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