Solve (frac{8sqrt{3}-sqrt{2}}{3})^2

$$( \frac { 8 \sqrt { 3 } - \sqrt { 2 } } { 3 } ) ^ { 2 }$$

Evaluate

$$\frac{194-16\sqrt{6}}{9}\approx 17.200907124$$

Solution Steps

To raise \frac{8\sqrt{3}-\sqrt{2}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8\sqrt{3}-\sqrt{2}\right)^{2}}{3^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8\sqrt{3}-\sqrt{2}\right)^{2}.

$$\frac{64\left(\sqrt{3}\right)^{2}-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{64\times 3-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

Multiply 64 and 3 to get 192.

$$\frac{192-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{192-16\sqrt{6}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{192-16\sqrt{6}+2}{3^{2}}$$

Add 192 and 2 to get 194.

$$\frac{194-16\sqrt{6}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{194-16\sqrt{6}}{9}$$

Expand

$$\frac{194 - 16 \sqrt{6}}{9} = 17.200907123941015$$

Solution Steps

To raise \frac{8\sqrt{3}-\sqrt{2}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8\sqrt{3}-\sqrt{2}\right)^{2}}{3^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8\sqrt{3}-\sqrt{2}\right)^{2}.

$$\frac{64\left(\sqrt{3}\right)^{2}-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{64\times 3-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

Multiply 64 and 3 to get 192.

$$\frac{192-16\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{192-16\sqrt{6}+\left(\sqrt{2}\right)^{2}}{3^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{192-16\sqrt{6}+2}{3^{2}}$$

Add 192 and 2 to get 194.

$$\frac{194-16\sqrt{6}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{194-16\sqrt{6}}{9}$$