Solve frac{sqrt{3}+sqrt{2}}{sqrt{65}-1}

$$\frac { \sqrt { 3 } + \sqrt { 2 } } { \sqrt { 65 } - 1 }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{3}+\sqrt{130}+\sqrt{195}}{64}\approx 0.445504042$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}+\sqrt{2}}{\sqrt{65}-1} by multiplying numerator and denominator by \sqrt{65}+1.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{65}+1\right)}{\left(\sqrt{65}-1\right)\left(\sqrt{65}+1\right)}$$

Consider \left(\sqrt{65}-1\right)\left(\sqrt{65}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{65}+1\right)}{\left(\sqrt{65}\right)^{2}-1^{2}}$$

Square \sqrt{65}. Square 1.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{65}+1\right)}{65-1}$$

Subtract 1 from 65 to get 64.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{65}+1\right)}{64}$$

Apply the distributive property by multiplying each term of \sqrt{3}+\sqrt{2} by each term of \sqrt{65}+1.

$$\frac{\sqrt{3}\sqrt{65}+\sqrt{3}+\sqrt{2}\sqrt{65}+\sqrt{2}}{64}$$

To multiply \sqrt{3} and \sqrt{65}, multiply the numbers under the square root.

$$\frac{\sqrt{195}+\sqrt{3}+\sqrt{2}\sqrt{65}+\sqrt{2}}{64}$$

To multiply \sqrt{2} and \sqrt{65}, multiply the numbers under the square root.

$$\frac{\sqrt{195}+\sqrt{3}+\sqrt{130}+\sqrt{2}}{64}$$