Solve 1/4+{2/3*[(frac{sqrt{3}+sqrt{4}}{(-2)})^2+(2*sqrt[3]{27})]-3/5+2/1-frac{3{5}}}

$$\frac { 1 } { 4 } + \{ \frac { 2 } { 3 } \cdot [ ( \frac { \sqrt { 3 } + \sqrt { 4 } } { ( - 2 ) } ) ^ { 2 } + ( 2 \cdot \sqrt[ 3 ] { 27 } ) ] - \frac { \frac { 3 } { 5 } + 2 } { 1 - \frac { 3 } { 5 } } \}$$

Evaluate

$$\frac{2\sqrt{3}}{3}-\frac{13}{12}\approx 0.071367205$$

Short Solution Steps

Calculate the square root of 4 and get 2.

$$\frac{1}{4}+\frac{2}{3}\left(\left(\frac{\sqrt{3}+2}{-2}\right)^{2}+2\sqrt[3]{27}\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Multiply both numerator and denominator by -1.

$$\frac{1}{4}+\frac{2}{3}\left(\left(\frac{-\sqrt{3}-2}{2}\right)^{2}+2\sqrt[3]{27}\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

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

$$\frac{1}{4}+\frac{2}{3}\left(\frac{\left(-\sqrt{3}-2\right)^{2}}{2^{2}}+2\sqrt[3]{27}\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Calculate \sqrt[3]{27} and get 3.

$$\frac{1}{4}+\frac{2}{3}\left(\frac{\left(-\sqrt{3}-2\right)^{2}}{2^{2}}+2\times 3\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Multiply 2 and 3 to get 6.

$$\frac{1}{4}+\frac{2}{3}\left(\frac{\left(-\sqrt{3}-2\right)^{2}}{2^{2}}+6\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6 times \frac{2^{2}}{2^{2}}.

$$\frac{1}{4}+\frac{2}{3}\left(\frac{\left(-\sqrt{3}-2\right)^{2}}{2^{2}}+\frac{6\times 2^{2}}{2^{2}}\right)-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Since \frac{\left(-\sqrt{3}-2\right)^{2}}{2^{2}} and \frac{6\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{1}{4}+\frac{2}{3}\times \frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2^{2}}-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Multiply \frac{2}{3} times \frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{4}+\frac{2\left(\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}\right)}{3\times 2^{2}}-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{1}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}-\frac{\frac{3}{5}+2}{1-\frac{3}{5}}$$

Add \frac{3}{5} and 2 to get \frac{13}{5}.

$$\frac{1}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}-\frac{\frac{13}{5}}{1-\frac{3}{5}}$$

Subtract \frac{3}{5} from 1 to get \frac{2}{5}.

$$\frac{1}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}-\frac{\frac{13}{5}}{\frac{2}{5}}$$

Divide \frac{13}{5} by \frac{2}{5} by multiplying \frac{13}{5} by the reciprocal of \frac{2}{5}.

$$\frac{1}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}-\frac{13}{5}\times \frac{5}{2}$$

Multiply \frac{13}{5} and \frac{5}{2} to get \frac{13}{2}.

$$\frac{1}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}-\frac{13}{2}$$

Subtract \frac{13}{2} from \frac{1}{4} to get -\frac{25}{4}.

$$-\frac{25}{4}+\frac{\left(-\sqrt{3}-2\right)^{2}+6\times 2^{2}}{2\times 3}$$

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

$$-\frac{25}{4}+\frac{\left(\sqrt{3}\right)^{2}+4\sqrt{3}+4+6\times 2^{2}}{2\times 3}$$

The square of \sqrt{3} is 3.

$$-\frac{25}{4}+\frac{3+4\sqrt{3}+4+6\times 2^{2}}{2\times 3}$$

Add 3 and 4 to get 7.

$$-\frac{25}{4}+\frac{7+4\sqrt{3}+6\times 2^{2}}{2\times 3}$$

Calculate 2 to the power of 2 and get 4.

$$-\frac{25}{4}+\frac{7+4\sqrt{3}+6\times 4}{2\times 3}$$

Multiply 6 and 4 to get 24.

$$-\frac{25}{4}+\frac{7+4\sqrt{3}+24}{2\times 3}$$

Add 7 and 24 to get 31.

$$-\frac{25}{4}+\frac{31+4\sqrt{3}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$-\frac{25}{4}+\frac{31+4\sqrt{3}}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 6 is 12. Multiply -\frac{25}{4} times \frac{3}{3}. Multiply \frac{31+4\sqrt{3}}{6} times \frac{2}{2}.

$$-\frac{25\times 3}{12}+\frac{2\left(31+4\sqrt{3}\right)}{12}$$

Since -\frac{25\times 3}{12} and \frac{2\left(31+4\sqrt{3}\right)}{12} have the same denominator, add them by adding their numerators.

$$\frac{-25\times 3+2\left(31+4\sqrt{3}\right)}{12}$$

Do the multiplications in -25\times 3+2\left(31+4\sqrt{3}\right).

$$\frac{-75+62+8\sqrt{3}}{12}$$

Do the calculations in -75+62+8\sqrt{3}.

$$\frac{-13+8\sqrt{3}}{12}$$

Factor

$$\frac{8 \sqrt{3} - 13}{12} = 0.07136720504591813$$