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

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

Evaluate

$$-\sqrt{6}-4\approx -6.449489743$$

Solution Steps

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

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

Consider \left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{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}.

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

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

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

Subtract 3 from 2 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

The square of \sqrt{2} is 2.

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

Subtract 2 from 6 to get 4.

$$-\left(2\sqrt{6}+4-\sqrt{2}\sqrt{3}\right)$$

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

$$-\left(2\sqrt{6}+4-\sqrt{6}\right)$$

Combine 2\sqrt{6} and -\sqrt{6} to get \sqrt{6}.

$$-\left(\sqrt{6}+4\right)$$

To find the opposite of \sqrt{6}+4, find the opposite of each term.

$$-\sqrt{6}-4$$

Factor

$$-\sqrt{6}-4$$