Solve frac{2sqrt{3}-4sqrt{2}}{sqrt{3}-sqrt{2}}=

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

Evaluate

$$-2\sqrt{6}-2\approx -6.898979486$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{3}-4\sqrt{2}}{\sqrt{3}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}+\sqrt{2}.

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

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}.

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

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

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 2\sqrt{3}-4\sqrt{2} by each term of \sqrt{3}+\sqrt{2}.

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

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

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

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

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

Combine 2\sqrt{6} and -4\sqrt{6} to get -2\sqrt{6}.

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

The square of \sqrt{2} is 2.

$$6-2\sqrt{6}-4\times 2$$

Multiply -4 and 2 to get -8.

$$6-2\sqrt{6}-8$$

Subtract 8 from 6 to get -2.

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

Factor

$$2 {(-\sqrt{6} - 1)} = -6.898979486$$