Formule Algebră

Proprietățile radicalilor; Logaritmi; Schimbarea bazei unui logaritm

Alte formule calcul prescurtat:

$$\begin{gathered} x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz -zx) \\ {x}^3 + y^3 + z^3 = {(x + y + z)}^3 - 3(x + y)(y + z)(z + x) \\ {a}^4 + b^4 = (a^2 + b^2 - ab\sqrt{2})(a^2 + b^2 + ab\sqrt{2}) \\ {a}^4 - b^4 = (a - b)(a + b)(a^2 + b^2) \end{gathered}$$ 

$$\begin{gathered} a^5 + b^5 = (a + b)(a^4 - a^{3}b + a^2{b}^2 - a{b}^3 + b^4) \\ {a}^5 - b^5 = (a - b)(a^4 + a^{3}b + a^2{b}^2 + a{b}^3 + b^4) \\ (1 + a)(1 + a^2 + a^4) = 1 + a + a^2 + a^3 + a^4 + a^5 \\ {a}^6 + b^6 = {(a^3 - 2a{b}^2)}^2 + {(b^3 - 2a^{2}b)}^2 \end{gathered}$$

$$\begin{gathered} a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + ... + a{b}^{n-2} + b^{n-1}) \\ {a}^{2n} - b^{2n} = (a^2 - b^2)(a^{2n-2} + a^{2n-4}{b}^2 + ... + a^2{b}^{2n-4} + b^{2n-2}) \\ {a}^{2n+2} + b^{2n+2} = (a + b)(a^{2n} + a^{2n-1}b + ... + a{b}^{2n-1} + b^{2n}) \\ (1 + a + a^2 + ... + a^n)(1 + a^{n+1}) = 1 + a + a^2 + a^3 + ... +a^{2n} + a^{2n+1} \\ \forall x, y ,z, a, b, c \in \mathrm{ℝ} și n \in \mathrm{\mathbb{N}} \end{gathered}$$

Proprietăţile radicalilor:

$$\begin{gathered} \sqrt[n]{a} = a^{\frac{1}{n}}, a > 0 \\ {\left(\sqrt[n]{\frac{1}{a}}\right)}^n = \frac{1}{a}, a > 0 \\ \sqrt[n]{a} ·\sqrt[n]{b} · \sqrt[n]{c} = \sqrt[n]{a·b·c}, (a, b, c \ge 0) \\ \sqrt[n]{a} : \sqrt[n]{b} = \sqrt[n]{\frac{a}{b}}, (a \ge 0, b > 0) \end{gathered}$$

$$\begin{gathered} \sqrt[m]{a}·\sqrt[n]{a} = \sqrt[mn]{a^{m+n}} \\ \sqrt[m]{\sqrt[n]{a}} = \sqrt[m·n]{a} = \sqrt[n]{\sqrt[m]{a}}, (a \ge 0) \\ \sqrt{a^2}= \left|a\right|, (a\in \mathrm{ℝ}) \end{gathered}$$

$$\begin{gathered} \sqrt[n]{a^{n·m}} = a^m, (a \ge 0) \\ \sqrt[m]{a^n} = {\left(\sqrt[m]{a}\right)}^n = a^{\frac{n}{m}}, \left(a \ge 0, m\ne 0\right) \\ \sqrt[m·n]{a^{m·p}} = \sqrt[n]{a^p}, (a > 0). \end{gathered}$$

Logaritmi

$$\begin{gathered} {\log }_a\left(a\right)=1 \\ {\log }_a\left(1\right)=0 \\ {\log }_a\left(a^c\right)=c \\ {\log }_a\left(\frac{1}{b}\right)=-{\log }_a\left(b\right) \\ {\log }_a\left(b\right) · {\log }_b\left(a\right)=1 \end{gathered}$$

$$\begin{gathered} {\log }_a\left(b^m\right) = m·{\log }_a\left(b\right) \\ {\log }_a\left(\sqrt[m]{b}\right) = \frac{1}{m}·{\log }_a\left(b\right), (b>0, m\in \mathrm{\mathbb{N}}, m\ge 2) \end{gathered}$$

$$\begin{gathered} {\log }_a\left(x\right) + {\log }_a\left(y\right) = {\log }_a\left(x·y\right), \forall x, y >0 \\ {\log }_a\left(\frac{x}{y}\right) = {\log }_a\left(x\right) - {\log }_a\left(y\right), \forall x, y >0 \end{gathered}$$

$\frac{{\log }_a\left(x\right)}{{\log }_a\left(y\right)}= \frac{{\log }_b\left(x\right)}{{\log }_b\left(y\right)}, \forall x, y, a, b >0, a\ne 1, b\ne 1.$

Schimbarea bazei unui logaritm

${\log }_a\left(b\right) = \frac{{\log }_c\left(b\right)}{{\log }_c\left(a\right)}.$

Alte cunoștințe utile despre logaritmi:

$\left.\begin{array}{r}0<a<1\\ 0<x<1\end{array}\right\}\Rightarrow {\log }_a\left(x\right) > 0$

$\left.\begin{array}{r}a>1\\ 0<x<1\end{array}\right\}\Rightarrow {\log }_a\left(x\right) < 0$

$\left.\begin{array}{r} a>1\\ x>1\end{array}\right\}\Rightarrow {\log }_a\left(x\right) > 0$

$\left.\begin{array}{r}0<a<1\\ x>1\end{array}\right\}\Rightarrow {\log }_a\left(x\right) < 0$

$\left.\begin{array}{r}a>1\\ 0<x<y\end{array}\right\}\Rightarrow {\log }_a\left(x\right) < {\log }_a\left(y\right)$

$\left.\begin{array}{r}0<a<1\\ 0<x<y\end{array}\right\}\Rightarrow {\log }_a\left(x\right) > {\log }_a\left(y\right)$

Și, mai mult, avem:

${\log }_a\left(x\right) = {\log }_{a^n}\left(x^n\right), x, a > 0, a\ne 1, n\in \mathbf{N}$

$a^x = e^{x\ln \left(a\right)}, x\in \mathbf{R} , a > 0, a\ne 1$

$\ln \left(a\right) = {\log }_e\left(a\right)$

$\log \left(a\right)={\log }_{10}\left(a\right).$

Combinatorică

Permutări

Reprezintă numărul tuturor submulțimilor unei mulțimi.

$P_n = n!$
                    

Aranjamente

Reprezintă numărul tuturor submulțimilor ordonate de $k$ elemente ale unei mulțimi de $n$ elemente.

$A_n^k = \frac{n!}{(n-k)!}$.

Combinări

Reprezintă numărul tuturor submulțimilor (neordonate) de $k$ elemente ale unei mulțimi de $n$ elemente.

$C_n^k = \frac{n!}{k! (n-k)!}.$

mai mult ...

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