Solve (-sqrt{5})^2+(sqrt{2}+1)(sqrt{2}-1)-sqrt{2/3}*sqrt{1}1/2

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

Evaluate

$$-\frac{\sqrt{6}}{6}+6\approx 5.59175171$$

Short Solution Steps

Calculate -\sqrt{5} to the power of 2 and get \left(\sqrt{5}\right)^{2}.

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

Consider \left(\sqrt{2}+1\right)\left(\sqrt{2}-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}. Square 1.

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

The square of \sqrt{2} is 2.

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

Subtract 1 from 2 to get 1.

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

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

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

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

Calculate the square root of 1 and get 1.

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

Multiply 1 and \frac{1}{2} to get \frac{1}{2}.

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

Multiply \frac{\sqrt{6}}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 3 and 2 to get 6.

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

The square of \sqrt{5} is 5.

$$5+1-\frac{\sqrt{6}}{6}$$

Add 5 and 1 to get 6.

$$6-\frac{\sqrt{6}}{6}$$

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

$$\frac{6\times 6}{6}-\frac{\sqrt{6}}{6}$$

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

$$\frac{6\times 6-\sqrt{6}}{6}$$

Do the multiplications in 6\times 6-\sqrt{6}.

$$\frac{36-\sqrt{6}}{6}$$

Factor

$$\frac{36 - \sqrt{6}}{6} = 5.591751709536137$$