Nota Add math Tingkatan 4 Bab 4

a^m × aⁿ = a^m+n; a^m ÷ aⁿ = a^m−n; (a^m)ⁿ = a^mn. (p.vii)
An algebraic expression involving indices is simplified using the laws of indices. (p.90)
If a^m = aⁿ then m = n; if a^m = b^m then a = b (a > 0, a ≠ 1). Express both sides in the same base. (p.92)

Rational numbers can be written as a/b (b ≠ 0), including recurring decimals. Irrational numbers cannot be converted to fractions. (p.96)
A surd is a number in root form that cannot be simplified to a rational number; surds have non-recurring infinite decimals. For example √2 is a surd but √4 = 2 is not. (p.97)
Recurring decimals are converted to fractions by multiplying by a suitable power of 10 and subtracting. (p.98)
For a > 0, b > 0: √a × √b = √(ab) and √a ÷ √b = √(a/b). (p.100)
Simplify surds by extracting the largest perfect square factor, e.g. √18 = 3√2. (p.101)
Surds with the same irrational factor are similar surds and can be added or subtracted. (p.103)
Rationalising the denominator: multiply numerator and denominator by the conjugate surd. Conjugate of m√a + n√b is m√a − n√b. (p.104)
√a × √a = a, and (a − √b)(a + √b) = a² − b. (p.104)

log_a N = x ⇔ N = a^x, where a > 0 and a ≠ 1. (p.110)
log_a 1 = 0 and log_a a = 1. log_a N is defined only if N > 0. (p.110)
Common logarithm is base 10 (log₁₀ a = lg a). If log₁₀ N = x then antilog x = N. (p.110)
If f : x → a^x then f⁻¹ : x → log_a x. (p.111)
Product law: log_a xy = log_a x + log_a y. (p.114)
Division law: log_a (x/y) = log_a x − log_a y. (p.114)
Power law: log_a xⁿ = n log_a x. (p.114)
Change of base: log_a b = log_c b / log_c a; in particular log_a b = 1 / log_b a. (p.117)
Natural logarithm: ln a = log_e a with e ≈ 2.7182. ln e = 1, ln e^x = x, e^{ln x} = x. (p.117)
Equations like 3^x = 70 that cannot be written in the same base are solved by taking logarithms of both sides. (p.119)

Growth/decay models (e.g. A = 1000 × 2^0.2n) are solved by substituting values or using logarithms to find the time. (p.122)

Chapter 4: Indices, Surds and Logarithms