The derivative of $f(x) = 5^{x^2}$ with respect to $x$ is:

B) $2x \cdot 5^{x^2} \cdot \ln 5$

The sum of the first n terms of an arithmetic sequence (S_n) is formulated as:

S_n = n / 2 * [ 2a + (n - 1) * d ] = n / 2 * [ a + l ]

The sum to infinity (S_infinity) of a geometric series exists and converges if and only if the common ratio satisfies:

|r| < 1 (i.e. -1 < r < 1), and S_infinity = a / (1 - r)

كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

Unit 1: Powers of Powers and Exponential Equations

Q: كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

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.

🌳 Concept tree

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

01 1-1 The base, the exponent, and the product and quotient of powers

In aⁿ, a is the base and n the exponent; aⁿ means a multiplied by itself n times.
aᵐ × aⁿ = aᵐ⁺ⁿ — same base multiplied: add the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ — same base divided: subtract the exponents (a ≠ 0).
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).

02 1-2 The zero exponent and the negative exponent

a⁰ = 1 for every a ≠ 0; 0⁰ is undefined. So (3x²)⁰ = 1 whenever x ≠ 0.
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.
(a/b)⁻ⁿ = (b/a)ⁿ.
Example: 2⁻³ = 1/8, and (2/5)⁻² = (5/2)² = 25/4.

03 1-3 Power of a power, of a product and of a quotient

(aᵐ)ⁿ = aᵐⁿ — multiply the exponents.
(ab)ⁿ = aⁿ bⁿ — the exponent distributes over a product.
(a/b)ⁿ = aⁿ/bⁿ — likewise over a quotient (b ≠ 0).
Nested powers: ((2³)²)⁴ = ²⁴ˣ⁴ = 2⁴⁸ — always multiply from the inside out.

04 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.

05 1-5 Powers, square roots and fractional exponents

a^1/n = ⁿ√a, and a^m/n = (ⁿ√a)ᵐ, defined for a ≥ 0 when n is even.
√(a²) = |a| — the absolute value, which is why √((−5)²) = 5 and not −5.
√(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, e.g. √48 = √(16 × 3) = 4√3.
Rationalising a denominator: 1/√2 = √2/2 and 1/(√3 + 1) = (√3 − 1)/2.

06 1-6 Simplifying expressions that combine several laws

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.
Example: (2x³y⁻²)³ / (4x²y)² = (8x⁹y⁻⁶)/(16x⁴y²) = ½ x⁵ y⁻⁸ = x⁵ / (2y⁸).
Leave the answer with positive exponents unless the question asks otherwise.

07 1-7 Exponential equations by equalising the base

If aˣ = aʸ with a > 0 and a ≠ 1, then x = y: write both sides as powers of the same base and equate the exponents.
Example: 3^x+1 = 27 → 3^x+1 = 3³ → x = 2.
Example: 4^x = 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.

08 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: x^3/2 = 8 → x = 8^2/3 = (∛8)² = 4.

09 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.

10 1-10 Equations in which the unknown is the base

xⁿ = a → x = a^1/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 → ∅.

Different bases

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.

11 1-11 Real-life applications

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.
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.
Attenuation: a quantity halving each period is (½)ⁿ of its original value.
Always state the unit and reject a non-integer or negative value of n when the situation is a count of periods.

12 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 ; a^m/n = (ⁿ√a)ᵐ.
(−a)ⁿ > 0 for even n, (−a)ⁿ < 0 for odd n.
aˣ = aʸ ⟹ x = y ; xⁿ = a ⟹ x = a^1/n (two signs for even n).

13 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: Numbers and Operations — Powers of Powers and Exponential Equations

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