Solve (frac{sqrt{4}}{2}-frac{sqrt{6}}{3})^2

$$( \frac { \sqrt { 4 } } { 2 } - \frac { \sqrt { 6 } } { 3 } ) ^ { 2 }$$

Evaluate

$$\frac{5-2\sqrt{6}}{3}\approx 0.033673505$$

Short Solution Steps

Calculate the square root of 4 and get 2.

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

Divide 2 by 2 to get 1.

$$\left(1-\frac{\sqrt{6}}{3}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

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

Since \frac{3}{3} and \frac{\sqrt{6}}{3} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{3-\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{6}\right)^{2}.

$$\frac{9-6\sqrt{6}+\left(\sqrt{6}\right)^{2}}{3^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{9-6\sqrt{6}+6}{3^{2}}$$

Add 9 and 6 to get 15.

$$\frac{15-6\sqrt{6}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{15-6\sqrt{6}}{9}$$

Expand

$$\frac{5 - 2 \sqrt{6}}{3} = 0.03367350481121475$$

Short Solution Steps

Calculate the square root of 4 and get 2.

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

Divide 2 by 2 to get 1.

$$\left(1-\frac{\sqrt{6}}{3}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

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

Since \frac{3}{3} and \frac{\sqrt{6}}{3} have the same denominator, subtract them by subtracting their numerators.

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

To raise \frac{3-\sqrt{6}}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{6}\right)^{2}.

$$\frac{9-6\sqrt{6}+\left(\sqrt{6}\right)^{2}}{3^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{9-6\sqrt{6}+6}{3^{2}}$$

Add 9 and 6 to get 15.

$$\frac{15-6\sqrt{6}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{15-6\sqrt{6}}{9}$$