MCQ
$\sqrt{11 \sqrt{11 \sqrt{11 \ldots .} 4 \text { terms }}}$ is equal to
- A$\sqrt[16]{11^5}$
- B$\sqrt[16]{11}$
- C$\sqrt[16]{11^{14}}$
- ✓$\sqrt[16]{11^{15}}$
✓
Answer
Correct option: D.
$\sqrt[16]{11^{15}}$
(d)
Let $x=\sqrt{11 \sqrt{11 \sqrt{11 \ldots .} 4 \text { terms }}}$. Then,
$x=\sqrt{11 \sqrt{11 \sqrt{11 \sqrt{11}}}}=\sqrt{11 \sqrt{11 \sqrt{11 \times 11^{1 / 2}}}}=\sqrt{11 \sqrt{11 \sqrt{11^{3 / 2}}}}=\sqrt{11 \sqrt{11 \times\left(11^{3 / 2}\right)^{1 / 2}}}$
$\Rightarrow x=\sqrt{11 \sqrt{11 \times 11^{3 / 4}}}=\sqrt{11 \sqrt{11^{7 / 4}}}=\sqrt{11 \times 11^{7 / 8}}=\sqrt{11^{1+\frac{7}{8}}}=\sqrt{11^{15 / 8}}=\left(11^{15 / 8}\right)^{1 / 2}$
$=(11)^{\frac{15}{8} \times \frac{1}{2}}=11^{15 / 16}=\sqrt[16]{11^{15}}$
Let $x=\sqrt{11 \sqrt{11 \sqrt{11 \ldots .} 4 \text { terms }}}$. Then,
$x=\sqrt{11 \sqrt{11 \sqrt{11 \sqrt{11}}}}=\sqrt{11 \sqrt{11 \sqrt{11 \times 11^{1 / 2}}}}=\sqrt{11 \sqrt{11 \sqrt{11^{3 / 2}}}}=\sqrt{11 \sqrt{11 \times\left(11^{3 / 2}\right)^{1 / 2}}}$
$\Rightarrow x=\sqrt{11 \sqrt{11 \times 11^{3 / 4}}}=\sqrt{11 \sqrt{11^{7 / 4}}}=\sqrt{11 \times 11^{7 / 8}}=\sqrt{11^{1+\frac{7}{8}}}=\sqrt{11^{15 / 8}}=\left(11^{15 / 8}\right)^{1 / 2}$
$=(11)^{\frac{15}{8} \times \frac{1}{2}}=11^{15 / 16}=\sqrt[16]{11^{15}}$
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