Comprehensive Mathematical Reference: Radicals, Exponents, Logarithms, Arithmetic Progressions, and Statistics

Properties of Radicals and Roots

• Multiplication property of radicals:   $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

• Addition of like radicals:   $a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c}$

• Subtraction of like radicals:   $a\sqrt{c} - b\sqrt{c} = (a - b)\sqrt{c}$

• Product of two square roots:   $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

• Quotient property of radicals:   $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

• Product of terms with numerical or variable coefficients and radicals:   $a\sqrt{c} \times b\sqrt{d} = ab\sqrt{cd}$

• Expansion of the square of a radical binomial (sum form):   $(\sqrt{a} + \sqrt{b})^2 = (a + b) + 2\sqrt{ab}$

• Expansion of the square of a radical binomial (difference form):   $(\sqrt{a} - \sqrt{b})^2 = (a + b) - 2\sqrt{ab}$

• Unnesting formula for square roots of radical expressions (sum form):   $\sqrt{(a + b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$

• Unnesting formula for square roots of radical expressions (difference form):   $\sqrt{(a + b) - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b}$

  • Condition: $a \ge b$

Properties of Exponents

• Definition of positive integer exponents (repeated multiplication):   $a^n = \underbrace{a \times a \times \dots \times a}_{n\text{ times}}$

• Zero exponent rule:   $a^0 = 1$

  • Restriction: $a \neq 0$

• Negative exponent rule (reciprocal representation):   $a^{-n} = \frac{1}{a^n}$

  • Restriction: $a \neq 0$

• Product rule of exponents (multiplication with identical bases):   $a^m \times a^n = a^{m+n}$

• Quotient rule of exponents (division with identical bases):   $\frac{a^m}{a^n} = a^{m-n}$

  • Restriction: $a \neq 0$

• Power of a product rule:   $(a \times b)^n = a^n \times b^n$

• Power of a quotient rule:   $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

  • Restriction: $b \neq 0$

• Power of a power rule:   $(a^m)^n = a^{m \times n}$

• Fractional exponent rule (conversion to radical expressions):   $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Properties and Laws of Logarithms

• Fundamental definition of a logarithm:   $a^c = b \iff {}^a\log(b) = c$

  • Base conditions: $a > 0$ and $a \neq 1$
  • Argument condition: $b > 0$

• Logarithm of 1:   ${}^a\log(1) = 0$

• Logarithm of the base:   ${}^a\log(a) = 1$

• Product law of logarithms:   ${}^a\log(b \times c) = {}^a\log(b) + {}^a\log(c)$

• Quotient law of logarithms:   ${}^a\log\left(\frac{b}{c}\right) = {}^a\log(b) - {}^a\log(c)$

• Power rule for the argument:   ${}^a\log(b^m) = m \times {}^a\log(b)$

• Power rule for both base and argument:   ${}^{a^n}\log(b^m) = \frac{m}{n} \times {}^a\log(b)$

• Power rule for the base:   ${}^{a^n}\log(b) = \frac{1}{n} \times {}^a\log(b)$

• Change of base formula using an arbitrary positive base $p$:   ${}^a\log(b) = \frac{{}^p\log(b)}{{}^p\log(a)}$

• Reciprocal base relation:   ${}^a\log(b) = \frac{1}{{}^b\log(a)}$

• Exponential-logarithmic identity:   $a^{{}^a\log(b)} = b$

• Chain rule / multiplication property of composite logarithms:   ${}^a\log(b) \times {}^b\log(c) \times {}^c\log(d) = {}^a\log(d)$

Arithmetic Sequences and Series

• Common difference ($b$) formula between adjacent terms:   $b = U_n - U_{n-1}$

• General formula for the $n$-th term ($U_n$) of an arithmetic sequence:   $U_n = a + (n - 1)b$

  • $a$: First term of the sequence ($a = U_1$
  • $b$: Common difference
  • $n$: Position index of the target term

• Formula for the sum of the first $n$ terms ($S_n$) using first term and common difference:   $S_n = \frac{n}{2}\left(2a + (n - 1)b\right)$

• Formula for the sum of the first $n$ terms ($S_n$) using first and last terms:   $S_n = \frac{n}{2}\left(a + U_n\right)$

• Formula for the middle term ($U_t$) in an arithmetic sequence with an odd number of terms:   $U_t = \frac{a + U_n}{2}$$U_t = \frac{U_1 + U_n}{2}$

  • Position index of the middle term: $t = \frac{n + 1}{2}$

• Formula to find the $n$-th term ($U_n$) using sequence partial sums:   $U_n = S_n - S_{n-1}$

Fundamentals of Statistics

• Statistical analysis involves the systematic collection, organization, presentation, analysis, and interpretation of numerical data sets.

• Measures of Central Tendency:

  • Mean: The arithmetic average, defined as the sum of all numerical values divided by the sample size $n$.
  • Median: The central observation value separating the upper half from the lower half of an ordered data array.
  • Mode: The specific value or values that appear with the highest frequency in a data set.

• Measures of Dispersion:

  • Range: The total numerical spread, calculated as the absolute difference between the maximum and minimum observations.
  • Variance: The average squared deviation of each quantitative value from the mean.
  • Standard Deviation: The square root of the variance, measuring absolute spread in original data units.

Sifat-Sifat Bentuk Akar dan Akar

• Sifat perkalian bentuk akar: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• Penjumlahan bentuk akar sejenis: $a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c}$
• Pengurangan bentuk akar sejenis: $a\sqrt{c} - b\sqrt{c} = (a - b)\sqrt{c}$
• Perkalian dua akar kuadrat: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$
• Sifat pembagian bentuk akar: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
• Perkalian suku dengan koefisien numerik atau variabel dan bentuk akar: $a\sqrt{c} \times b\sqrt{d} = ab\sqrt{cd}$
• Pengembangan kuadrat dari binomial bentuk akar (bentuk penjumlahan): $(\sqrt{a} + \sqrt{b})^2 = (a + b) + 2\sqrt{ab}$
• Pengembangan kuadrat dari binomial bentuk akar (bentuk pengurangan): $(\sqrt{a} - \sqrt{b})^2 = (a + b) - 2\sqrt{ab}$
• Rumus menyederhanakan akar dalam akar (bentuk penjumlahan): $\sqrt{(a + b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$
• Rumus menyederhanakan akar dalam akar (bentuk pengurangan): $\sqrt{(a + b) - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b}$
• Syarat: $a \ge b$

Sifat-Sifat Eksponen

• Definisi eksponen bulat positif (perkalian berulang): $a^n = \underbrace{a \times a \times \dots \times a}_{n\text{ kali}}$
• Aturan eksponen nol: $a^0 = 1$
  • Syarat: $a \neq 0$
• Aturan eksponen negatif (pangkat negatif): $a^{-n} = \frac{1}{a^n}$
  • Syarat: $a \neq 0$
• Sifat perkalian eksponen (perkalian dengan basis sama): $a^m \times a^n = a^{m+n}$
• Sifat pembagian eksponen (pembagian dengan basis sama): $\frac{a^m}{a^n} = a^{m-n}$
  • Syarat: $a \neq 0$
• Sifat perpangkatan dari perkalian: $(a \times b)^n = a^n \times b^n$
• Sifat perpangkatan dari pembagian: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
  • Syarat: $b \neq 0$
• Sifat pemangkatan eksponen: $(a^m)^n = a^{m \times n}$
• Sifat eksponen pecahan (konversi ke bentuk akar): $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Sifat-Sifat dan Hukum Logaritma

• Definisi dasar logaritma: $a^c = b \iff {}^a\log(b) = c$
  • Syarat basis: $a > 0$ dan $a \neq 1$
  • Syarat numerus: $b > 0$
• Logaritma dari 1: ${}^a\log(1) = 0$
• Logaritma dari basisnya sendiri: ${}^a\log(a) = 1$
• Sifat penjumlahan/perkalian logaritma: ${}^a\log(b \times c) = {}^a\log(b) + {}^a\log(c)$
• Sifat pengurangan/pembagian logaritma: ${}^a\log\left(\frac{b}{c}\right) = {}^a\log(b) - {}^a\log(c)$
• Sifat pangkat pada numerus: ${}^a\log(b^m) = m \times {}^a\log(b)$
• Sifat pangkat pada basis dan numerus: ${}^{a^n}\log(b^m) = \frac{m}{n} \times {}^a\log(b)$
• Sifat pangkat pada basis: ${}^{a^n}\log(b) = \frac{1}{n} \times {}^a\log(b)$
• Rumus mengubah basis logaritma dengan basis positif sembarang $p$: ${}^a\log(b) = \frac{{}^p\log(b)}{{}^p\log(a)}$
• Hubungan kebalikan basis logaritma: ${}^a\log(b) = \frac{1}{{}^b\log(a)}$
• Identitas eksponen-logaritma: $a^{{}^a\log(b)} = b$
• Sifat perkalian / aturan rantai logaritma: ${}^a\log(b) \times {}^b\log(c) \times {}^c\log(d) = {}^a\log(d)$

Barisan dan Deret Aritmatika

• Rumus beda ($b$) antara suku-suku berurutan: $b = U<em>n - U</em>{n-1}$
• Rumus umum suku ke-$n$ ($U<em>n$) dari barisan aritmatika: $U</em>n = a + (n - 1)b$
  • $a$: Suku pertama dari barisan ($a = U_1$
  • $b$: Beda
  • $n$: Indeks posisi suku yang dicari
• Rumus jumlah $n$ suku pertama ($S<em>n$) menggunakan suku pertama dan beda: $S</em>n = \frac{n}{2}\left(2a + (n - 1)b\right)$
• Rumus jumlah $n$ suku pertama ($S<em>n$) menggunakan suku pertama dan suku terakhir: $S</em>n = \frac{n}{2}\left(a + U_n\right)$
• Rumus suku tengah ($U<em>t$) pada barisan aritmatika dengan jumlah suku ganjil: $U</em>t = \frac{a + U<em>n}{2}$, $U</em>t = \frac{U<em>1 + U</em>n}{2}$
  • Indeks posisi suku tengah: $t = \frac{n + 1}{2}$
• Rumus mencari suku ke-$n$ ($U<em>n$) menggunakan jumlah parsial deret: $U</em>n = S<em>n - S</em>{n-1}$

Dasar-Dasar Statistika

• Analisis statistik melibatkan pengumpulan, pengorganisasian, penyajian, analisis, dan interpretasi himpunan data numerik secara sistematis.
• Ukuran Pemusatan Data:
  • Mean (Rata-rata): Rata-rata hitung, didefinisikan sebagai jumlah semua nilai numerik dibagi dengan ukuran sampel $n$.
  • Median: Nilai pengamatan tengah yang memisahkan separuh atas dari separuh bawah array data terurut.
  • Modus: Nilai spesifik yang paling sering muncul dalam suatu himpunan data.
• Ukuran Penyebaran Data:
  • Jangkauan (Range): Total rentang numerik, dihitung sebagai selisih mutlak antara nilai pengamatan maksimum dan minimum.
  • Varians: Rata-rata kuadrat deviasi dari setiap nilai kuantitatif terhadap rata-rata.
  • Simpangan Baku (Standard Deviation): Akar kuadrat dari varians, mengukur penyebaran mutlak dalam satuan data asli.