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

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

Evaluate

$$\frac{3\left(\sqrt{3}+2\right)}{8}\approx 1.399519053$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

$$\frac{9+6\sqrt{3}+3}{4^{2}}$$

Add 9 and 3 to get 12.

$$\frac{12+6\sqrt{3}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{12+6\sqrt{3}}{16}$$

Expand

$$\frac{3 \sqrt{3}}{8} + \frac{3}{4} = 1.399519053$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

$$\frac{9+6\sqrt{3}+3}{4^{2}}$$

Add 9 and 3 to get 12.

$$\frac{12+6\sqrt{3}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{12+6\sqrt{3}}{16}$$