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

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

Evaluate

$$2\sqrt{6}+5\approx 9.898979486$$

Solution Steps

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

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

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

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

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}.

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

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

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

Subtract 2 from 3 to get 1.

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

Anything divided by one gives itself.

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

Multiply \sqrt{3}+\sqrt{2} and \sqrt{3}+\sqrt{2} to get \left(\sqrt{3}+\sqrt{2}\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+\sqrt{2}\right)^{2}.

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

The square of \sqrt{3} is 3.

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

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

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

The square of \sqrt{2} is 2.

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

Add 3 and 2 to get 5.

$$\left(5+2\sqrt{6}\right)\times 1$$

Use the distributive property to multiply 5+2\sqrt{6} by 1.

$$5+2\sqrt{6}$$

Factor

$$2 \sqrt{6} + 5 = 9.898979486$$