| If \[x=\sqrt{a\sqrt[3]{b\sqrt{a\sqrt[3]{b.....\infty }}}},\]then the value of x is [SSC (CGL) Mains 2015] |
A) \[\sqrt[5]{a{{b}^{3}}}\]
B) \[\sqrt[5]{{{a}^{5}}b}\]
C) \[\sqrt[3]{{{a}^{3}}b}\]
D) \[\sqrt[5]{{{a}^{3}}b}\]
Correct Answer: D
Solution :
| Given, \[x=\sqrt{a\sqrt[3]{b\sqrt{a\sqrt[3]{b.....\infty }}}}\] |
| By squaring both sides, |
| \[{{x}^{2}}=a\sqrt[3]{b\sqrt{a\sqrt[3]{b.....\infty }}}\] |
| By cubing both sides, |
| \[{{x}^{6}}={{a}^{3}}b\sqrt{a\sqrt[3]{b.....\infty }}\] |
| \[\Rightarrow \] \[{{x}^{6}}={{a}^{3}}bx\] |
| \[\Rightarrow \] \[{{x}^{6}}-{{a}^{3}}bx=0\] |
| \[\Rightarrow \] \[x\,({{x}^{5}}-{{a}^{3}}b)=0\] |
| \[\Rightarrow \] \[x=0,\]\[{{x}^{5}}={{a}^{3}}b\] |
| \[\therefore \] \[x=\sqrt[5]{{{a}^{3}}b}\] |
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