Puteri, radicali, logaritmi. Reguli de calcul.

Reguli de calcul cu puteri

Pentru \displaystyle a,b\in \left ( 0,\, +\infty \right ) şi \displaystyle m,n\in \mathbb{R} :

\displaystyle a^{1}=a

\displaystyle a^{0}=1

\displaystyle 0^{0} nu are sens

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

\displaystyle a^{m} : a^{n}=a^{m-n}

\displaystyle \left (a^{m} \right )^{n}=a^{m\cdot n}

\displaystyle a^{m^{n}}=a^{\left ( m^{n} \right )}

\displaystyle a^{m}\cdot b^{m}=\left ( a\cdot b \right )^{m}

\displaystyle a^{m}:b^{m}=\left ( a:b \right )^{m}

\displaystyle \frac{a^{m}}{b^{m}}=\left ( \frac{a}{b} \right )^{m}

\displaystyle a^{-n}=\frac{1}{a^{n}}

\displaystyle \left ( \frac{a}{b} \right )^{-n}=\left ( \frac{b}{a} \right )^{n}

\displaystyle \left ( -1 \right )^{n}=\left\{\begin{matrix} 1,\: \textrm{dac} \breve{\textrm{a}}\: n=\textrm{par}\\ \\ -1,\: \textrm{dac} \breve{\textrm{a}}\: n=\textrm{impar} \end{matrix}\right.

\displaystyle \left ( -a \right )^{2k}=a^{2k},\: k\in \mathbb{Z}

\displaystyle \left ( -a \right )^{2k+1}=-a^{2k+1},\: k\in \mathbb{Z}

Puteri cu exponent raţional

Dacă \displaystyle a>0 şi \displaystyle m,n\in \mathbb{N}^{*},\: n\geq 2

\displaystyle a^{\frac{m}{n}}=\sqrt[n]{a^{m}}

Reguli de calcul cu radicali

Pentru \displaystyle a,b\in \left ( 0,\, +\infty \right ) şi \displaystyle m,n,p\in \mathbb{N}^{*},\: m\geq 2,\: n\geq 2 :

\displaystyle \sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}

\displaystyle \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}

\displaystyle \left (\sqrt[n]{a} \right )^{m}=\sqrt[n]{a^{m}}

\displaystyle \sqrt[n]{a^{n}\cdot b}=a\sqrt[n]{b}

\displaystyle \sqrt[n]{a^{m}}=\sqrt[np]{a^{mp}}

\displaystyle \sqrt[m]{\sqrt[n]{a}}=\sqrt[m\cdot n]{a}

Pentru \displaystyle n impar, \displaystyle \sqrt[n]{\left ( -a \right )}=-\sqrt[n]{a}

Formulele radicalilor suprapuşi

\displaystyle \sqrt{a+\sqrt{b}}=\sqrt{\frac{a+c}{2}}+\sqrt{\frac{a-c}{2}}

\displaystyle \sqrt{a-\sqrt{b}}=\sqrt{\frac{a+c}{2}}-\sqrt{\frac{a-c}{2}}

\displaystyle c=\sqrt{a^{2}-b} .

Reguli de calcul cu logaritmi

Fie numerele \displaystyle a,x,y\in \left ( 0,\, +\infty \right ),\: a\neq 1

\displaystyle \log _{a}x+\log _{a}y=\log _{a}\left (x\cdot y \right )

\displaystyle \log _{a}x-\log _{a}y=\log _{a}\left (\frac{x}{y} \right )

\displaystyle \log _{a}x^{k}=k\cdot \log _{a}x

\displaystyle \log _{a}x=\frac{\log _{b}x}{\log _{b}a},\; b\neq 1  (formula de schimbare a bazei)