Evaluate: sqrt(2\sqrt{2)}

Expression: $\sqrt{2\sqrt{2}}$

$\sqrt{\sqrt{2}}=\sqrt[4]{2}$

$=\sqrt{2}\sqrt[4]{2}$

Apply radical rule $\sqrt{a}=a^{\frac{1}{2}}$

$=2^{\frac{1}{2}}\sqrt[4]{2}$

Apply radical rule $\sqrt[n]{a}=a^{\frac{1}{n}}$

$=2^{\frac{1}{2}}\cdot 2^{\frac{1}{4}}$

Apply exponent rule $a^b\cdot a^c=a^{b+c}$

$=2^{\frac{1}{2}+\frac{1}{4}}$

Apply radical rule $\sqrt{ab}=\sqrt{a}\sqrt{b},\quad a\ge0, b\ge0$

$=\sqrt{2}\sqrt{\sqrt{2}}$

$2^{\frac{3}{4}}=\sqrt[4]{2^{3}}$

$=\sqrt[4]{2^{3}}$