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

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

Evaluate

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

Solution Steps

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

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

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}.

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

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

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

Subtract 3 from 2 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply -2 and 3 to get -6.

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

Subtract 6 from 2 to get -4.

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

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

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

The opposite of -4 is 4.

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

The opposite of -\sqrt{6} is \sqrt{6}.

$$4+\sqrt{6}$$