Solve [(sqrt{2})^{2}*(2/sqrt{3})^2][(1/2)^2+4*1^2-2^2]

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

Evaluate

$$\frac{2}{3}\approx 0.666666667$$

Solution Steps

The square of \sqrt{2} is 2.

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

Rationalize the denominator of \frac{2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To raise \frac{2\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Calculate 1 to the power of 2 and get 1.

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

Multiply 4 and 1 to get 4.

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

Add \frac{1}{4} and 4 to get \frac{17}{4}.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{17}{4}-2^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{17}{4}-4\right)$$

Subtract 4 from \frac{17}{4} to get \frac{1}{4}.

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

Multiply 2 and \frac{1}{4} to get \frac{1}{2}.

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

Expand \left(2\sqrt{3}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$\frac{1}{2}\times \frac{4\times 3}{3^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{1}{2}\times \frac{12}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{2}\times \frac{12}{9}$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}\times \frac{4}{3}$$

Multiply \frac{1}{2} and \frac{4}{3} to get \frac{2}{3}.

$$\frac{2}{3}$$

Factor

$$\frac{2}{3} = 0.6666666666666666$$