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

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

Evaluate

$$2\sqrt{3}+3-4\sqrt{2}\approx 0.807247366$$

Solution Steps

Rationalize the denominator of \frac{2+\sqrt{5}}{\sqrt{3}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{2}.

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

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

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

Square \sqrt{3}. Square \sqrt{2}.

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply 2+\sqrt{5} by \sqrt{3}-\sqrt{2}.

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

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

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{5}\sqrt{2}-\sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)+\left(1-\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{10}-\sqrt{5}\left(\sqrt{3}-\sqrt{2}\right)+\left(1-\sqrt{2}\right)^{2}$$

Use the distributive property to multiply \sqrt{5} by \sqrt{3}-\sqrt{2}.

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{10}-\left(\sqrt{5}\sqrt{3}-\sqrt{5}\sqrt{2}\right)+\left(1-\sqrt{2}\right)^{2}$$

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

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{10}-\left(\sqrt{15}-\sqrt{5}\sqrt{2}\right)+\left(1-\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{10}-\left(\sqrt{15}-\sqrt{10}\right)+\left(1-\sqrt{2}\right)^{2}$$

To find the opposite of \sqrt{15}-\sqrt{10}, find the opposite of each term.

$$2\sqrt{3}-2\sqrt{2}+\sqrt{15}-\sqrt{10}-\sqrt{15}+\sqrt{10}+\left(1-\sqrt{2}\right)^{2}$$

Combine \sqrt{15} and -\sqrt{15} to get 0.

$$2\sqrt{3}-2\sqrt{2}-\sqrt{10}+\sqrt{10}+\left(1-\sqrt{2}\right)^{2}$$

Combine -\sqrt{10} and \sqrt{10} to get 0.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\sqrt{2}\right)^{2}.

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

The square of \sqrt{2} is 2.

$$2\sqrt{3}-2\sqrt{2}+1-2\sqrt{2}+2$$

Add 1 and 2 to get 3.

$$2\sqrt{3}-2\sqrt{2}+3-2\sqrt{2}$$

Combine -2\sqrt{2} and -2\sqrt{2} to get -4\sqrt{2}.

$$2\sqrt{3}-4\sqrt{2}+3$$