Solve frac{sqrt{5}-sqrt{3}}{sqrt{2}-1}*frac{sqrt{5}+sqrt{3}}{sqrt{5}+sqrt{3}}

$$\frac { \sqrt { 5 } - \sqrt { 3 } } { \sqrt { 2 } - 1 } \times \frac { \sqrt { 5 } + \sqrt { 3 } } { \sqrt { 5 } + \sqrt { 3 } }$$

Evaluate

$$\sqrt{5}+\sqrt{10}-\sqrt{3}-\sqrt{6}\approx 1.216805087$$

Solution Steps

Divide \sqrt{5}+\sqrt{3} by \sqrt{5}+\sqrt{3} to get 1.

$$\frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}-1}\times 1$$

Rationalize the denominator of \frac{\sqrt{5}-\sqrt{3}}{\sqrt{2}-1} by multiplying numerator and denominator by \sqrt{2}+1.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}\times 1$$

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+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}.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}\times 1$$

Square \sqrt{2}. Square 1.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{2}+1\right)}{2-1}\times 1$$

Subtract 1 from 2 to get 1.

$$\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{2}+1\right)}{1}\times 1$$

Anything divided by one gives itself.

$$\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{2}+1\right)\times 1$$

Apply the distributive property by multiplying each term of \sqrt{5}-\sqrt{3} by each term of \sqrt{2}+1.

$$\left(\sqrt{5}\sqrt{2}+\sqrt{5}-\sqrt{3}\sqrt{2}-\sqrt{3}\right)\times 1$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\left(\sqrt{10}+\sqrt{5}-\sqrt{3}\sqrt{2}-\sqrt{3}\right)\times 1$$

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

$$\left(\sqrt{10}+\sqrt{5}-\sqrt{6}-\sqrt{3}\right)\times 1$$

Use the distributive property to multiply \sqrt{10}+\sqrt{5}-\sqrt{6}-\sqrt{3} by 1.

$$\sqrt{10}+\sqrt{5}-\sqrt{6}-\sqrt{3}$$