Question

Solve the following:
${(\sqrt{{(\sqrt{2}-\frac{3}{2})}^2}-\sqrt[3]{3(1-\sqrt{2}{)}^3})}^2+2^{-\frac{3}{2}}\cos \frac{3\pi }{4}$

Answer 1

${(\sqrt{{(\sqrt{2}-\frac{3}{2})}^2}-\sqrt[3]{3(1-\sqrt{2}{)}^3})}^2+2^{-\frac{3}{2}}\cos \frac{3\pi }{4}$

$={({\left({(\sqrt{2}-\frac{3}{2})}^2\right)}^{\frac{1}{2}}-((1-\sqrt{2}{)}^3{)}^{\frac{1}{3}})}^2$ $+2^{\frac{-3}{2}}\cos \frac{\pi }{4}$

$={(\sqrt{2}-\frac{3}{2}-(1-\sqrt{2}\left)\right)}^2$ $-2^{\frac{-3}{2}}\times \sqrt{2}$

$={(2\sqrt{2}-\frac{5}{2})}^2$ $-2^{\frac{3}{2}-\frac{1}{2}}$

$=8+\frac{25}{4}-10\sqrt{2}-2$

$=\frac{57}{4}-\sqrt{2}(10-\sqrt{2})$