Question

If x= ³√7+5 ^√2 - ____1___
³√7+5 ^√2

then find the value of x³+3x-14

Answer 1

Answer :

Given : x = $\sqrt[3]{7}+5\sqrt{2}-\frac{1}{\sqrt[3]{7}+5\sqrt{2}}$
So,

$$\begin{gathered} \Rightarrow \sqrt[3]{7}+5\sqrt{2}-\frac{1}{\sqrt[3]{7}+5\sqrt{2}}\times \frac{\sqrt[3]{7}-5\sqrt{2}}{\sqrt[3]{7}-5\sqrt{2}} \\ \Rightarrow \sqrt[3]{7}+5\sqrt{2}-\frac{\sqrt[3]{7}-5\sqrt{2}}{7-50} \\ \Rightarrow \sqrt[3]{7}+5\sqrt{2}-\frac{\sqrt[3]{7}-5\sqrt{2}}{-43} \\ \Rightarrow \sqrt[3]{7}+5\sqrt{2}+\frac{\sqrt[3]{7}-5\sqrt{2}}{43} \\ \Rightarrow \frac{43\sqrt[3]{7}+215\sqrt{2}+\sqrt[3]{7}-5\sqrt{2}}{43} \\ \Rightarrow \frac{44\sqrt[3]{7}+210\sqrt{2}}{43} \\ So, \\ \mathit{x}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{44}\mathbf{}\sqrt[\mathbf{3}\mathbf{}]{\mathbf{}\mathbf{7}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{210}\mathbf{}\sqrt{\mathbf{2}}\mathbf{}\mathbf{}}{\mathbf{43}}\mathbf{} \end{gathered}$$

Substitute that value in x³ + 3x - 14 , and get


$$\begin{gathered} \Rightarrow {\left(\frac{44\sqrt[3]{7}+210\sqrt{2}}{43}\right)}^3+3\left(\frac{44\sqrt[3]{7}+210\sqrt{2}}{43}\right)-14 \\ \Rightarrow {\left(\frac{44\sqrt[3]{7}}{43}+\frac{210\sqrt{2}}{43}\right)}^3+3\left(\frac{44\sqrt[3]{7}+210\sqrt{2}}{43}\right)-14 \\ \Rightarrow \left(\frac{596288}{79507}+\frac{18522000\sqrt{2}}{79507}+27720\times \sqrt[3]{7}\times \sqrt{2}+\left(\frac{44\sqrt[3]{7}}{43}+\frac{210\sqrt{2}}{43}\right)\right)+3\left(\frac{44\sqrt[3]{7}+210\sqrt{2}}{43}\right)-14(Asweknow(a\hspace{0.17em}+\hspace{0.17em}b{)}^3=a^3+b^3+3ab(a+b)) \\ \Rightarrow \left(\frac{596288}{79507}+\frac{18522000\sqrt{2}}{79507}+\frac{8537760\sqrt{2}}{43}+\frac{11642400\sqrt[3]{7}}{43}\right)+\frac{132\sqrt[3]{7}+630\sqrt{2}}{43}-14 \\ \Rightarrow \left(\frac{596288}{79507}+\frac{18522000\sqrt{2}}{79507}+\frac{15786318240\sqrt{2}}{79507}+\frac{11642400\sqrt[3]{7}}{43}\right)+\frac{132\sqrt[3]{7}}{43}+\frac{630\sqrt{2}}{43}-14 \\ \Rightarrow \frac{596288}{79507}-\frac{1113098}{79507}+\frac{18522000\sqrt{2}}{79507}+\frac{15786318240\sqrt{2}}{79507}\frac{1164870\sqrt{2}}{79507}+\frac{11642400\sqrt[3]{7}}{43}+\frac{132\sqrt[3]{7}}{43} \\ \mathbf{\Rightarrow }\mathbf{-}\frac{\mathbf{516810}}{\mathbf{79507}}\mathbf{}\mathbf{+}\frac{\mathbf{15806005110}\sqrt{\mathbf{2}}\mathbf{}\mathbf{}}{\mathbf{79507}}\mathbf{+}\frac{\mathbf{11642532}\mathbf{}\sqrt[\mathbf{3}\mathbf{}]{\mathbf{}\mathbf{7}}\mathbf{}}{\mathbf{43}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{}\mathit{A}\mathit{n}\mathit{s}\mathbf{}\mathbf{)} \end{gathered}$$