Question

E7-e,f,g
Este urgent!!!

Answer 1

Răspuns:

E7.

e) $(\sqrt[4]{625}+\sqrt[4]{3})(5+\sqrt[4]{48}-\sqrt[4]{243})\cdot(\sqrt{4+\sqrt{12}}+24)$

• Determini Radacinile: $\sqrt[4]{625}=5$
• Scoti factori de sub radical: $\sqrt[4]{48} = 2\sqrt[4]{3}$ , $\sqrt[4]{243} = 3\sqrt[4]{3}$ si $\sqrt{12} = 2\sqrt{3}$ .

$=(5+\sqrt[4]{3})\cdot(5+2\sqrt[4]{3}-3\sqrt[4]{3})\cdot(\sqrt{4+2\sqrt{3}}+24)$

• Reduci termenii asemenea (scazand coeficientii acestora): $2+2\sqrt[4]{3}-3\sqrt[4]{3} = (2=3)\sqrt[4]{3} = -1\sqrt[4]{3} = -\sqrt[4]{3}$
• Foloseste formula $\boxed{a^2+2ab+b^2=(a+b)^2}$ pentru: $\sqrt{4+2\sqrt{3}} = 1+2\sqrt{3}+3 = (1+\sqrt{3})^2$

$=(5+\sqrt[4]{3})\cdot(5-\sqrt[4]{3})\cdot(\sqrt{(1+\sqrt{3})^2}+24)$

• Foloseste formula $\boxed{(a-b)(a+b)=a^2-b^2}$ pentru: $(5+\sqrt[4]{3})\cdot(5-\sqrt[4]{3})=5^2-\sqrt[4]{3}^2 =25-\sqrt{3}$
• Aminteste-ti ca un radical fara ordin, are ordinul implicit 2: $\boxed{\sqrt{a}=\sqrt[2]{a}}$
• Simplifica ordinul radicalului si exponentul de sub radical cu 2: $\sqrt{(1+\sqrt{3}}^2= \sqrt[2]{(1+\sqrt{3}}^2=1+\sqrt{3}$

$=(25-\sqrt{3})\cdot(1+\sqrt{3}+24)$

• Observam ca putem aduna numerele din a doua paranteza: $(1+\sqrt{3}+24) = (25+\sqrt{3})$

$=(25-\sqrt{3})\cdot(25+\sqrt{3})$

• Folosim din nou formula amintita: $\boxed{(a-b)(a+b)=a^2-b^2}$

$=625-3$

• Efectuam scaderea numerelor:

$\boxed{=622}$

Tu cand vei scrie, le vei scrie doar pe cele cu = inainte, cele cu bulinuta sunt explicarile.

f) $\sqrt[3]{\sqrt[4]{27}}\cdot\sqrt[4]{\sqrt[3]{81}}\cdot\sqrt[6]{9\sqrt{3}}$

$=\sqrt[12]{27}\cdot\sqrt[12]{81}\cdot\sqrt[12]{(9\sqrt{3})^2}$

$=\sqrt[12]{27\cdot81\cdot(9\sqrt{3})^2}$

$=\sqrt[12]{27\cdot81\cdot81\cdot3}$

$=\sqrt[12]{531441}$

$\boxed{=3}$

g) $\sqrt[2]{2\sqrt[5]{2}}\cdot\sqrt[10]{4\sqrt[4]{2^5\cdot8}}$

$=\sqrt[10]{(2\sqrt[5]{2})^5}\cdot\sqrt[10]{2^2\cdot\sqrt[4]{2^5\cdot2^3}}$

$=\sqrt[10]{(2\sqrt[5]{2})^5}\cdot\sqrt[10]{2^2\cdot\sqrt[4]{2^8}}$

$=\sqrt[10]{(2\sqrt[5]{2})^5}\cdot\sqrt[10]{(2\cdot2)^2}$

$=\sqrt[10]{(2\sqrt[5]{2})^5\cdot(2\cdot2)^2}$

$=\sqrt[10]{32\cdot2\cdot4\cdot4}$

$=\sqrt[10]{1024}$

$\boxed{=2}$

Sunt de parere ca s-ar fi putut rezolva si in alta modalitate, insa eu asa as fi rezolvat-o.

Sper ca te-am ajutat!