Solve {left(frac{3/4right)}^2sqrt{1/8}}{sqrt[4]{{left(3/4right)}^4}}

$$\frac{ { \left( \frac{ 3 }{ 4 } \right) }^{ 2 } \sqrt{ \frac{ 1 }{ 8 } } }{ \sqrt[ 4 ]{ { \left( \frac{ 3 }{ 4 } \right) }^{ 4 } } }$$

Evaluate

$$\frac{3\sqrt{2}}{16}\approx 0.265165043$$

Solution Steps

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\frac{\frac{9}{16}\sqrt{\frac{1}{8}}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{\frac{9}{16}\times \frac{\sqrt{1}}{\sqrt{8}}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{9}{16}\times \frac{1}{\sqrt{8}}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\frac{9}{16}\times \frac{1}{2\sqrt{2}}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{9}{16}\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{9}{16}\times \frac{\sqrt{2}}{2\times 2}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{9}{16}\times \frac{\sqrt{2}}{4}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Multiply \frac{9}{16} times \frac{\sqrt{2}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{9\sqrt{2}}{16\times 4}}{\sqrt[4]{\left(\frac{3}{4}\right)^{4}}}$$

Calculate \frac{3}{4} to the power of 4 and get \frac{81}{256}.

$$\frac{\frac{9\sqrt{2}}{16\times 4}}{\sqrt[4]{\frac{81}{256}}}$$

Calculate \sqrt[4]{\frac{81}{256}} and get \frac{3}{4}.

$$\frac{\frac{9\sqrt{2}}{16\times 4}}{\frac{3}{4}}$$

Divide \frac{9\sqrt{2}}{16\times 4} by \frac{3}{4} by multiplying \frac{9\sqrt{2}}{16\times 4} by the reciprocal of \frac{3}{4}.

$$\frac{9\sqrt{2}\times 4}{16\times 4\times 3}$$

Cancel out 3\times 4 in both numerator and denominator.

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

Multiply 4 and 4 to get 16.

$$\frac{3\sqrt{2}}{16}$$