Математика, страница 2

$t=A:p$

$1\,\text{дм}^2=100\,\text{см}^2,\quad 1\,\text{м}^2=100\,\text{дм}^2=10000\,\text{см}^2$

$S=S_1+S_2+\dots+S_n$

$S=S_{\text{большая}}-S_{\text{вырез}}$

$S=a\cdot b,\quad P=2(a+b)$

$N=a_n\cdot10^n+a_{n-1}\cdot10^{n-1}+\dots+a_1\cdot10+a_0$

$\bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$

$\frac{m}{n}=m\cdot\frac{1}{n}$

$\text{часть}=A\cdot\frac{m}{n}=A:n\cdot m$

$A=\text{часть}:m\cdot n$

$p\%=\frac{p}{100}$

$\text{часть}=A\cdot\frac{p}{100}$

$S=a\cdot b$

$V=a\cdot b\cdot c$

$S_{\text{пов}}=2(ab+bc+ac)$

$n\vdots2\Leftrightarrow a_0\in\{0,2,4,6,8\},\quad n\vdots5\Leftrightarrow a_0\in\{0,5\},\quad n\vdots10\Leftrightarrow a_0=0$

$n\vdots3\Leftrightarrow S(n)\vdots3,\quad n\vdots9\Leftrightarrow S(n)\vdots9$

$p>1,\;D(p)=\{1,p\}$

$n=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_k^{\alpha_k}$

$\gcd(a,b)=\prod p_i^{\min(\alpha_i,\beta_i)}$

$\operatorname{lcm}(a,b)=\prod p_i^{\max(\alpha_i,\beta_i)}$

$\frac{a}{b}=\frac{a:d}{b:d},\quad d=\gcd(a,b)$

$\frac{a}{m}=\frac{a\cdot(k/m)}{k},\quad \frac{b}{n}=\frac{b\cdot(k/n)}{k},\quad k=\operatorname{lcm}(m,n)$

$\frac{a}{m}\pm\frac{b}{n}=\frac{a\cdot(k/m)\pm b\cdot(k/n)}{k},\quad k=\operatorname{lcm}(m,n)$

$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d},\quad b\ne0,\;d\ne0$

$x = -\frac{b}{a},\quad a \ne 0$

$\frac{a}{b} = \frac{c}{d}\quad \Longleftrightarrow \quad ad = bc$

$a^m \cdot a^n = a^{m+n}$

$\frac{a^m}{a^n} = a^{m-n},\quad a \ne 0$

$(ab)^n = a^n b^n$

$(a^m)^n = a^{mn}$

$y = kx + b$

$k = \frac{y_2 - y_1}{x_2 - x_1},\quad x_1 \ne x_2$

$\alpha + \beta = 180^\circ$

$\alpha = \beta$

$\alpha + \beta + \gamma = 180^\circ$

$\alpha_{\text{внеш}} = \beta + \gamma$

$P = a + b + c$

$P = 2(a + b)$

$S = ab$

$d = |x_2 - x_1|$

$ka + ma = (k + m)a$

$(ax^m)(bx^n) = abx^{m+n}$

$(ax^m)^n = a^n x^{mn}$

$a(b + c) = ab + ac$

$(a + b)(c + d) = ac + ad + bc + bd$

$ab + ac = a(b + c)$

$ax + ay + bx + by = (a + b)(x + y)$

$ax + by = c$

$y = kx + b,\quad ax + by = c$

$a_1x + b_1y = c_1,\quad a_2x + b_2y = c_2$

$ax + b = c,\quad x = \frac{c - b}{a},\quad a \ne 0$

$A = B \Longleftrightarrow A + m = B + m$

$y = kx$

$k = \frac{y_2 - y_1}{x_2 - x_1},\quad y - y_1 = k(x - x_1)$

$\sqrt{a}=x\quad\Longleftrightarrow\quad x^2=a,\;x\ge 0,\;a\ge 0$

$(\sqrt{a})^2=a,\quad a\ge 0$

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}},\quad a\ge 0,\;b>0$

$\sqrt{a^2b}=|a|\sqrt{b},\quad b\ge 0$

$a\sqrt{b}=\sqrt{a^2b}\quad\text{при }a\ge 0,\;b\ge 0$