Solve +H:[(frac{2-sqrt{3}}{2})^2+2/sqrt{3+1}]*1/sqrt{6}

$$+ H : [ ( \frac { 2 - \sqrt { 3 } } { 2 } ) ^ { 2 } + \frac { 2 } { \sqrt { 3 } + 1 } ] \cdot \frac { 1 } { \sqrt { 6 } }$$

Evaluate

$$\frac{2\sqrt{6}H}{9}$$

Short Solution Steps

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

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

Rationalize the denominator of \frac{2}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

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

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3}-1 times \frac{2^{2}}{2^{2}}.

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

Since \frac{\left(2-\sqrt{3}\right)^{2}}{2^{2}} and \frac{\left(\sqrt{3}-1\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Divide H by \frac{\left(2-\sqrt{3}\right)^{2}+\left(\sqrt{3}-1\right)\times 2^{2}}{2^{2}} by multiplying H by the reciprocal of \frac{\left(2-\sqrt{3}\right)^{2}+\left(\sqrt{3}-1\right)\times 2^{2}}{2^{2}}.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{H\times 4}{\left(2-\sqrt{3}\right)^{2}+\left(\sqrt{3}-1\right)\times 2^{2}}\times \frac{1}{\sqrt{6}}$$

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

$$\frac{H\times 4}{4-4\sqrt{3}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{3}-1\right)\times 2^{2}}\times \frac{1}{\sqrt{6}}$$

The square of \sqrt{3} is 3.

$$\frac{H\times 4}{4-4\sqrt{3}+3+\left(\sqrt{3}-1\right)\times 2^{2}}\times \frac{1}{\sqrt{6}}$$

Add 4 and 3 to get 7.

$$\frac{H\times 4}{7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 2^{2}}\times \frac{1}{\sqrt{6}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{H\times 4}{7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 4}\times \frac{1}{\sqrt{6}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{H\times 4}{7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 4}\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{H\times 4}{7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 4}\times \frac{\sqrt{6}}{6}$$

Multiply \frac{H\times 4}{7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 4} times \frac{\sqrt{6}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{H\times 4\sqrt{6}}{\left(7-4\sqrt{3}+\left(\sqrt{3}-1\right)\times 4\right)\times 6}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\sqrt{6}H}{3\left(4\left(\sqrt{3}-1\right)-4\sqrt{3}+7\right)}$$

Use the distributive property to multiply 4 by \sqrt{3}-1.

$$\frac{2\sqrt{6}H}{3\left(4\sqrt{3}-4-4\sqrt{3}+7\right)}$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$\frac{2\sqrt{6}H}{3\left(-4+7\right)}$$

Add -4 and 7 to get 3.

$$\frac{2\sqrt{6}H}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{2\sqrt{6}H}{9}$$

Differentiate w.r.t. H

$$\frac{2 \sqrt{6}}{9} = 0.5443310539518174$$