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

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

Evaluate

$$2\sqrt{3}+3\sqrt{2}\approx 7.706742302$$

Solution Steps

Divide \sqrt{3}+\sqrt{2} by \sqrt{3}+\sqrt{2} to get 1.

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

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

$$\frac{\sqrt{6}}{\sqrt{3}-\sqrt{2}}\times 1$$

Rationalize the denominator of \frac{\sqrt{6}}{\sqrt{3}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}+\sqrt{2}.

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

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}.

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

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

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply \sqrt{6} by \sqrt{3}+\sqrt{2}.

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

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Use the distributive property to multiply 3\sqrt{2}+2\sqrt{3} by 1.

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