Question

sa se aduca la o forma mai simpla:
a)
$(1 + \sqrt{3} )(1 + \sqrt[4]{3} )(1 + \sqrt[8]{3} )$
b)
$(1 + \sqrt[3]{2} + \sqrt[3]{4})(1 + \sqrt[9]{2} + \sqrt[9]{4} )$

​

Answer 1

Răspuns:

$a)~(1+\sqrt{3})(1+\sqrt[4]{3})(1+\sqrt[8]{3})=\frac{(1+\sqrt{3})(1+\sqrt[4]{3})(1+\sqrt[8]{3})(1-\sqrt[8]{3}) }{1-\sqrt[8]{3} }= \frac{(1+\sqrt{3})(1+\sqrt[4]{3})(1^{2}-(\sqrt[8]{3})^{2}) }{1-\sqrt[8]{3} }= \frac{(1+\sqrt{3})(1+\sqrt[4]{3})(1-\sqrt[4]{3}) }{1-\sqrt[8]{3} }= \frac{(1+\sqrt{3})(1^{2}-(\sqrt[4]{3})^{2}) }{1-\sqrt[8]{3} }=\frac{(1+\sqrt{3})(1-\sqrt{3}) }{1-\sqrt[8]{3} }=\frac{1^{2}-(\sqrt{3})^{2} }{1-\sqrt[8]{3} } =\frac{1-3}{1-\sqrt[8]{3} }=\frac{-2}{1-\sqrt[8]{3} } =$

$=\frac{2}{\sqrt[8]{3}-1 }.$

Explicație pas cu pas:

$b)~\frac{(1+\sqrt[3]{2}+\sqrt[3]{4})(1+\sqrt[9]{2}+\sqrt[9]{4} )(1-\sqrt[9]{2})}{1-\sqrt[9]{2}}= \frac{(1+\sqrt[3]{2}+\sqrt[3]{4})(1^{3}-(\sqrt[9]{2})^{3})}{1-\sqrt[9]{2}}= \frac{(1+\sqrt[3]{2}+\sqrt[3]{4})(1-\sqrt[9]{2^{3}})}{1-\sqrt[9]{2}}= \frac{(1+\sqrt[3]{2}+\sqrt[3]{4})(1-\sqrt[3]{2})}{1-\sqrt[9]{2}}=\frac{1^{3}-(\sqrt[3]{2})^{3} }{1-\sqrt[9]{2} } =\frac{1-2}{1-\sqrt[9]{2}} =\frac{-1}{1-\sqrt[9]{2}}=\frac{1}{\sqrt[9]{2}-1}$