Solve frac{sqrt{15}+sqrt{6}+sqrt{5}+sqrt{2}}{sqrt{3}+1}-frac{sqrt{15}-sqrt{10}-sqrt{6}+2}{sqrt{3}-sqrt{2}}

$$\frac { \sqrt { 15 } + \sqrt { 6 } + \sqrt { 5 } + \sqrt { 2 } } { \sqrt { 3 } + 1 } - \frac { \sqrt { 15 } - \sqrt { 10 } - \sqrt { 6 } + 2 } { \sqrt { 3 } - \sqrt { 2 } }$$

Evaluate

$$2\sqrt{2}\approx 2.828427125$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}}{\sqrt{3}+1} by multiplying numerator and denominator by \sqrt{3}-1.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}-\frac{\sqrt{15}-\sqrt{10}-\sqrt{6}+2}{\sqrt{3}-\sqrt{2}}$$

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-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}.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}-\frac{\sqrt{15}-\sqrt{10}-\sqrt{6}+2}{\sqrt{3}-\sqrt{2}}$$

Square \sqrt{3}. Square 1.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{3-1}-\frac{\sqrt{15}-\sqrt{10}-\sqrt{6}+2}{\sqrt{3}-\sqrt{2}}$$

Subtract 1 from 3 to get 2.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\frac{\sqrt{15}-\sqrt{10}-\sqrt{6}+2}{\sqrt{3}-\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{15}-\sqrt{10}-\sqrt{6}+2}{\sqrt{3}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}+\sqrt{2}.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\frac{\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}$$

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}.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\frac{\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\frac{\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)}{3-2}$$

Subtract 2 from 3 to get 1.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\frac{\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)}{1}$$

Anything divided by one gives itself.

$$\frac{\left(\sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2}\right)\left(\sqrt{3}-1\right)}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of \sqrt{15}+\sqrt{6}+\sqrt{5}+\sqrt{2} by each term of \sqrt{3}-1.

$$\frac{\sqrt{15}\sqrt{3}-\sqrt{15}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{5}\sqrt{3}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}\sqrt{3}-\sqrt{15}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{5}\sqrt{3}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

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

$$\frac{3\sqrt{5}-\sqrt{15}+\sqrt{6}\sqrt{3}-\sqrt{6}+\sqrt{5}\sqrt{3}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

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}.

$$\frac{3\sqrt{5}-\sqrt{15}+\sqrt{3}\sqrt{2}\sqrt{3}-\sqrt{6}+\sqrt{5}\sqrt{3}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

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

$$\frac{3\sqrt{5}-\sqrt{15}+3\sqrt{2}-\sqrt{6}+\sqrt{5}\sqrt{3}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{3\sqrt{5}-\sqrt{15}+3\sqrt{2}-\sqrt{6}+\sqrt{15}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Combine -\sqrt{15} and \sqrt{15} to get 0.

$$\frac{3\sqrt{5}+3\sqrt{2}-\sqrt{6}-\sqrt{5}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Combine 3\sqrt{5} and -\sqrt{5} to get 2\sqrt{5}.

$$\frac{2\sqrt{5}+3\sqrt{2}-\sqrt{6}+\sqrt{2}\sqrt{3}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

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

$$\frac{2\sqrt{5}+3\sqrt{2}-\sqrt{6}+\sqrt{6}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Combine -\sqrt{6} and \sqrt{6} to get 0.

$$\frac{2\sqrt{5}+3\sqrt{2}-\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Combine 3\sqrt{2} and -\sqrt{2} to get 2\sqrt{2}.

$$\frac{2\sqrt{5}+2\sqrt{2}}{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Divide each term of 2\sqrt{5}+2\sqrt{2} by 2 to get \sqrt{5}+\sqrt{2}.

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{15}-\sqrt{10}-\sqrt{6}+2\right)\left(\sqrt{3}+\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of \sqrt{15}-\sqrt{10}-\sqrt{6}+2 by each term of \sqrt{3}+\sqrt{2}.

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{15}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{10}\sqrt{3}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{3}\sqrt{5}\sqrt{3}+\sqrt{15}\sqrt{2}-\sqrt{10}\sqrt{3}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

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

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}+\sqrt{15}\sqrt{2}-\sqrt{10}\sqrt{3}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}+\sqrt{30}-\sqrt{10}\sqrt{3}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}+\sqrt{30}-\sqrt{30}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

Combine \sqrt{30} and -\sqrt{30} to get 0.

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}-\sqrt{10}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}-\sqrt{2}\sqrt{5}\sqrt{2}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

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

$$\sqrt{5}+\sqrt{2}-\left(3\sqrt{5}-2\sqrt{5}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

Combine 3\sqrt{5} and -2\sqrt{5} to get \sqrt{5}.

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{5}-\sqrt{6}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

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}.

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{5}-\sqrt{3}\sqrt{2}\sqrt{3}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

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

$$\sqrt{5}+\sqrt{2}-\left(\sqrt{5}-3\sqrt{2}-\sqrt{6}\sqrt{2}+2\sqrt{3}+2\sqrt{2}\right)$$

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}.

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

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

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

Combine -2\sqrt{3} and 2\sqrt{3} to get 0.

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

Combine -3\sqrt{2} and 2\sqrt{2} to get -\sqrt{2}.

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

To find the opposite of \sqrt{5}-\sqrt{2}, find the opposite of each term.

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

The opposite of -\sqrt{2} is \sqrt{2}.

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

Combine \sqrt{5} and -\sqrt{5} to get 0.

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

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

$$2\sqrt{2}$$