Proprietatile radicalilor

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$$ 1. \sqrt[n]{a^m} = a^{m\over n} $$ $$ 2. \sqrt[n]{a_1}*\sqrt[n]{a_2}*...*\sqrt[n]{a_k} = \sqrt[n]{a_1*a_2*...*a_k} $$ $$ 3. (\sqrt[n]{a})^m = \sqrt[n]{a^m} $$ $$ 4. {\sqrt[n]{a}\over\sqrt[n]{b}} = \sqrt[n]{a\over b} $$ $$ 5. \sqrt[n]{a} * \sqrt[m]{a} = \sqrt[nm]{a^{n+m}} $$ $$ 6. \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} = \sqrt[m]{\sqrt[n]{a}} $$

$$ 7. \sqrt[n]{a^p} * \sqrt[m]{b^k} = \sqrt[nm]{a^{pm} * b^{kn}} $$ $$ 8. {\sqrt[n]{a^p}\over\sqrt[m]{b^k}} = \sqrt[nm]{a^{pm}\over b^{kn}} $$ $$ 9. (\sqrt[n]{a})^n = |a| $$ $$ 10. \sqrt[2n+1]{-a} = -\sqrt[2n+1]{a} $$ $$ 11. \sqrt{a} \pm \sqrt{b} = \sqrt{a\pm 2\sqrt{ab} + b} $$

$$ 12. Scoaterea\hspace{3pt} factorilor\hspace{3pt} de\hspace{3pt} sub\hspace{3pt} radical : \sqrt[n]{a^{mn}*b^k} = a^m * \sqrt[n]{b^k} $$ $$ 13. Introducerea\hspace{3pt} factorilor\hspace{3pt} sub\hspace{3pt} radical : a^m * \sqrt[n]{b^k} = \sqrt[n]{a^{mn}*b^k} $$ $$ 14. "Amplificarea" radicalilor : \sqrt[n]{a^p} = \sqrt[nm]{a^{pm}} $$ $$ 15. Formula\hspace{3pt} radicalilor\hspace{3pt} compusi : \sqrt{A \pm \sqrt{B}} = \sqrt{A+C\over 2} \pm \sqrt{A-C\over 2} , unde\hspace{3pt} C = \sqrt{A^2 - B} $$ $$ 16. Pentru\hspace{3pt} \sqrt{a} \pm \sqrt{b}\hspace{3pt} expresia\hspace{3pt} conjugata\hspace{3pt} este : \sqrt{a} \mp \sqrt{b} $$ $$ 17. Pentru\hspace{3pt} \sqrt[3]{a} \pm \sqrt[3]{b}\hspace{3pt} expresia\hspace{3pt} conjugata\hspace{3pt} este : \sqrt[3]{b^2} \mp \sqrt[3]{ab} + \sqrt[3]{b^2} $$
$$ \underline{Conditii\hspace{3pt} de\hspace{3pt} existenta} : $$ $$ 1. Pentru\hspace{3pt} radicali\hspace{3pt} de\hspace{3pt} ordin\hspace{3pt} par,\hspace{3pt} de\hspace{3pt} forma\hspace{3pt} \sqrt[2n]{E(x)},\hspace{3pt} se\hspace{3pt} va\hspace{3pt} pune,\hspace{3pt} obligatoriu,\hspace{3pt} $$ $$ conditia\hspace{3pt} E(x)>=0 $$ $$ 2. Pentru\hspace{3pt} radicali\hspace{3pt} de\hspace{3pt} ordin\hspace{3pt} impar,\hspace{3pt} de\hspace{3pt} forma\hspace{3pt} \sqrt[2n+1]{E(x)},\hspace{3pt} nu\hspace{3pt} se\hspace{3pt} pun\hspace{3pt} conditii\hspace{3pt} de\hspace{3pt} existenta\hspace{3pt} si, $$ $$ ca\hspace{3pt} urmare,\hspace{3pt} x\hspace{3pt} poate\hspace{3pt} fi\hspace{3pt} orice\hspace{3pt} numar\hspace{3pt} real. $$