| If \[a={{(\sqrt{2}+1)}^{-1/3}},\] then find out the value of \[\left( {{a}^{3}}-\frac{1}{{{a}^{3}}} \right).\] |
A) 0
B) \[-2\sqrt{2}\]
C) \[3\sqrt{2}\]
D) \[-2\]
Correct Answer: D
Solution :
| \[a={{(\sqrt{2}+1)}^{-1/3}}\]\[\Rightarrow \]\[a={{\left( \frac{1}{\sqrt{2}+1} \right)}^{1/3}}\] |
| \[{{a}^{3}}=\frac{1}{\sqrt{2}+1}=\frac{1}{\sqrt{2}+1}\times \frac{(\sqrt{2}-1)}{(\sqrt{2}+1)}=\frac{\sqrt{2}-1}{1}=\sqrt{2}-1\]\[\therefore \] \[\frac{1}{{{a}^{3}}}=\frac{1}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}\,\,=\,\,\sqrt{2}+1\] |
| So, \[{{a}^{3}}-\frac{1}{{{a}^{3}}}=\sqrt{2}-1-(\sqrt{2}+1)\] |
| \[=\sqrt{2}-1-\sqrt{2}-1=-\,2\] |
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