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

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

Evaluate

$$-\frac{3\sqrt{3}}{4}+\frac{3}{2}\approx 0.200961894$$

Solution Steps

To raise \frac{3-\sqrt{3}}{4} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{\left(3-\sqrt{3}\right)^{2}}{4^{2}} as a single fraction.

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

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

$$\frac{2\left(9-6\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)}{4^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\left(9-6\sqrt{3}+3\right)}{4^{2}}$$

Add 9 and 3 to get 12.

$$\frac{2\left(12-6\sqrt{3}\right)}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{2\left(12-6\sqrt{3}\right)}{16}$$

Divide 2\left(12-6\sqrt{3}\right) by 16 to get \frac{1}{8}\left(12-6\sqrt{3}\right).

$$\frac{1}{8}\left(12-6\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{8} by 12-6\sqrt{3}.

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

Expand

$$-\frac{3 \sqrt{3}}{4} + \frac{3}{2} = 0.200961894$$

Solution Steps

To raise \frac{3-\sqrt{3}}{4} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{\left(3-\sqrt{3}\right)^{2}}{4^{2}} as a single fraction.

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

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

$$\frac{2\left(9-6\sqrt{3}+\left(\sqrt{3}\right)^{2}\right)}{4^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\left(9-6\sqrt{3}+3\right)}{4^{2}}$$

Add 9 and 3 to get 12.

$$\frac{2\left(12-6\sqrt{3}\right)}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{2\left(12-6\sqrt{3}\right)}{16}$$

Divide 2\left(12-6\sqrt{3}\right) by 16 to get \frac{1}{8}\left(12-6\sqrt{3}\right).

$$\frac{1}{8}\left(12-6\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{8} by 12-6\sqrt{3}.

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