Solve (1/2)+4(frac{2/sqrt{3})-(1)^2}{(1/2)^2+(frac{sqrt{3}}{2})^2}

$$\frac { ( \frac { 1 } { 2 } ) + 4 ( \frac { 2 } { \sqrt { 3 } } ) - ( 1 ) ^ { 2 } } { ( \frac { 1 } { 2 } ) ^ { 2 } + ( \frac { \sqrt { 3 } } { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{8\sqrt{3}}{3}-\frac{1}{2}\approx 4.118802154$$

Short Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Express 4\times \frac{2\sqrt{3}}{3} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{1}{2} times \frac{3}{3}. Multiply \frac{4\times 2\sqrt{3}}{3} times \frac{2}{2}.

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

Since \frac{3}{6} and \frac{2\times 4\times 2\sqrt{3}}{6} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3+2\times 4\times 2\sqrt{3}.

$$\frac{\frac{3+16\sqrt{3}}{6}-1^{2}}{\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\frac{3+16\sqrt{3}}{6}-1}{\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{6}{6}.

$$\frac{\frac{3+16\sqrt{3}}{6}-\frac{6}{6}}{\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

Since \frac{3+16\sqrt{3}}{6} and \frac{6}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3+16\sqrt{3}-6}{6}}{\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

Do the calculations in 3+16\sqrt{3}-6.

$$\frac{\frac{-3+16\sqrt{3}}{6}}{\left(\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\frac{-3+16\sqrt{3}}{6}}{\frac{1}{4}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$$

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

$$\frac{\frac{-3+16\sqrt{3}}{6}}{\frac{1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\frac{-3+16\sqrt{3}}{6}}{\frac{1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{4}}$$

Since \frac{1}{4} and \frac{\left(\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-3+16\sqrt{3}}{6}}{\frac{1+\left(\sqrt{3}\right)^{2}}{4}}$$

Divide \frac{-3+16\sqrt{3}}{6} by \frac{1+\left(\sqrt{3}\right)^{2}}{4} by multiplying \frac{-3+16\sqrt{3}}{6} by the reciprocal of \frac{1+\left(\sqrt{3}\right)^{2}}{4}.

$$\frac{\left(-3+16\sqrt{3}\right)\times 4}{6\left(1+\left(\sqrt{3}\right)^{2}\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\left(16\sqrt{3}-3\right)}{3\left(\left(\sqrt{3}\right)^{2}+1\right)}$$

The square of \sqrt{3} is 3.

$$\frac{2\left(16\sqrt{3}-3\right)}{3\left(3+1\right)}$$

Add 3 and 1 to get 4.

$$\frac{2\left(16\sqrt{3}-3\right)}{3\times 4}$$

Multiply 3 and 4 to get 12.

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

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

$$\frac{1}{6}\left(16\sqrt{3}-3\right)$$

Use the distributive property to multiply \frac{1}{6} by 16\sqrt{3}-3.

$$\frac{8}{3}\sqrt{3}-\frac{1}{2}$$

Factor

$$\frac{16 \sqrt{3} - 3}{6} = 4.118802153517006$$