Solve frac{2+sqrt{3}}{1-sqrt{2}}+frac{1-sqrt{2}}{2+sqrt{3}}

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

Evaluate

$$-2\sqrt{3}-4\sqrt{2}\approx -9.120955865$$

Solution Steps

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

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

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

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

Square 1. Square \sqrt{2}.

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

Subtract 2 from 1 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

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

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 2+\sqrt{3} by each term of 1+\sqrt{2}.

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

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

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

To find the opposite of 2+2\sqrt{2}+\sqrt{3}+\sqrt{6}, find the opposite of each term.

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

Apply the distributive property by multiplying each term of 1-\sqrt{2} by each term of 2-\sqrt{3}.

$$-2-2\sqrt{2}-\sqrt{3}-\sqrt{6}+2-\sqrt{3}-2\sqrt{2}+\sqrt{3}\sqrt{2}$$

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

$$-2-2\sqrt{2}-\sqrt{3}-\sqrt{6}+2-\sqrt{3}-2\sqrt{2}+\sqrt{6}$$

Add -2 and 2 to get 0.

$$-2\sqrt{2}-\sqrt{3}-\sqrt{6}-\sqrt{3}-2\sqrt{2}+\sqrt{6}$$

Combine -\sqrt{3} and -\sqrt{3} to get -2\sqrt{3}.

$$-2\sqrt{2}-2\sqrt{3}-\sqrt{6}-2\sqrt{2}+\sqrt{6}$$

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

$$-4\sqrt{2}-2\sqrt{3}-\sqrt{6}+\sqrt{6}$$

Combine -\sqrt{6} and \sqrt{6} to get 0.

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

Factor

$$2 {(-\sqrt{3} - 2 \sqrt{2})} = -9.120955865$$