Question

The value of $\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}$ is

• A. ${\left(2\right)}^{\frac{1}{2}}$ `
• B. ${\left(2\right)}^{\frac{1}{4}}$
• C. ${\left(2\right)}^{\frac{1}{8}}$
• D. ${\left(2\right)}^{\frac{15}{16}}$

Answer 1

Answer: (D)

${\left(2\right)}^{\frac{15}{16}}$

$\sqrt{2}$$=2^{\frac{1}{2}}$
and power of common base is added like $2^a\times 2^b=2^{a+b}$

$\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}=$$\sqrt{2\sqrt{2\sqrt{2\times 2^{\frac{1}{2}}}}}$

$=\sqrt{2\sqrt{2\sqrt{2^{\frac{3}{2}}}}}$

$=\sqrt{2\sqrt{2\times 2^{\frac{3}{4}}}}$

$=\sqrt{2\sqrt{2^{\frac{7}{4}}}}$

$=\sqrt{2\times 2^{\frac{7}{8}}}$

$=\sqrt{2^{\frac{15}{8}}}$
$=2^{\frac{15}{16}}$