Solve {left(sqrt{sqrt[3]{1/27}}right)}^2-{left(1/3div2/3*frac{1{9}}-1right)}^-3+0.5*sqrt{100}

$${ \left( \sqrt{ \sqrt[3]{ \frac{ 1 }{ 27 } } } \right) }^{ 2 } - { \left( \frac{ \frac{ 1 }{ 3 } \div 2 }{ 3 \times \frac{ 1 }{ 9 } } -1 \right) }^{ -3 } +0.5 \times \sqrt{ 100 }$$

Evaluate

$$\frac{40}{3}\approx 13.333333333$$

Solution Steps

Calculate \sqrt[3]{\frac{1}{27}} and get \frac{1}{3}.

$$\left(\sqrt{\frac{1}{3}}\right)^{2}-\left(\frac{\frac{\frac{1}{3}}{2}}{3\times \frac{1}{9}}-1\right)^{-3}+0.5\sqrt{100}$$

The square of \sqrt{\frac{1}{3}} is \frac{1}{3}.

$$\frac{1}{3}-\left(\frac{\frac{\frac{1}{3}}{2}}{3\times \frac{1}{9}}-1\right)^{-3}+0.5\sqrt{100}$$

Express \frac{\frac{1}{3}}{2} as a single fraction.

$$\frac{1}{3}-\left(\frac{\frac{1}{3\times 2}}{3\times \frac{1}{9}}-1\right)^{-3}+0.5\sqrt{100}$$

Multiply 3 and 2 to get 6.

$$\frac{1}{3}-\left(\frac{\frac{1}{6}}{3\times \frac{1}{9}}-1\right)^{-3}+0.5\sqrt{100}$$

Multiply 3 and \frac{1}{9} to get \frac{1}{3}.

$$\frac{1}{3}-\left(\frac{\frac{1}{6}}{\frac{1}{3}}-1\right)^{-3}+0.5\sqrt{100}$$

Divide \frac{1}{6} by \frac{1}{3} by multiplying \frac{1}{6} by the reciprocal of \frac{1}{3}.

$$\frac{1}{3}-\left(\frac{1}{6}\times 3-1\right)^{-3}+0.5\sqrt{100}$$

Multiply \frac{1}{6} and 3 to get \frac{1}{2}.

$$\frac{1}{3}-\left(\frac{1}{2}-1\right)^{-3}+0.5\sqrt{100}$$

Subtract 1 from \frac{1}{2} to get -\frac{1}{2}.

$$\frac{1}{3}-\left(-\frac{1}{2}\right)^{-3}+0.5\sqrt{100}$$

Calculate -\frac{1}{2} to the power of -3 and get -8.

$$\frac{1}{3}-\left(-8\right)+0.5\sqrt{100}$$

The opposite of -8 is 8.

$$\frac{1}{3}+8+0.5\sqrt{100}$$

Add \frac{1}{3} and 8 to get \frac{25}{3}.

$$\frac{25}{3}+0.5\sqrt{100}$$

Calculate the square root of 100 and get 10.

$$\frac{25}{3}+0.5\times 10$$

Multiply 0.5 and 10 to get 5.

$$\frac{25}{3}+5$$

Add \frac{25}{3} and 5 to get \frac{40}{3}.

$$\frac{40}{3}$$

Factor

$$\frac{5 \cdot 2 ^ {3}}{3} = 13\frac{1}{3} = 13.333333333333334$$