Solve (frac{sqrt{5}-sqrt{3}}{2})^2

$$( \frac { \sqrt { 5 } - \sqrt { 3 } } { 2 } ) ^ { 2 }$$

Evaluate

$$-\frac{\sqrt{15}}{2}+2\approx 0.063508327$$

Solution Steps

To raise \frac{\sqrt{5}-\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{3}\right)^{2}.

$$\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5-2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5-2\sqrt{15}+3}{2^{2}}$$

Add 5 and 3 to get 8.

$$\frac{8-2\sqrt{15}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8-2\sqrt{15}}{4}$$

Expand

$$-\frac{\sqrt{15}}{2} + 2 = 0.063508327$$

Solution Steps

To raise \frac{\sqrt{5}-\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{3}\right)^{2}.

$$\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5-2\sqrt{5}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5-2\sqrt{15}+\left(\sqrt{3}\right)^{2}}{2^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5-2\sqrt{15}+3}{2^{2}}$$

Add 5 and 3 to get 8.

$$\frac{8-2\sqrt{15}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{8-2\sqrt{15}}{4}$$