Solve frac{6+3sqrt{10}}{sqrt{2}+sqrt{3}+sqrt{5}}*frac{sqrt{2}+sqrt{3}+sqrt{5}}{sqrt{2}+sqrt{3+48}}

$$\frac { 6 + 3 \sqrt { 10 } } { \sqrt { 2 } + \sqrt { 3 } + \sqrt { 5 } } \times \frac { \sqrt { 2 } + \sqrt { 3 } + \sqrt { 5 } } { \sqrt { 2 } + \sqrt { 3 + 48 } }$$

Evaluate

$$\frac{3\left(\sqrt{510}+2\sqrt{51}-2\sqrt{2}-2\sqrt{5}\right)}{49}\approx 1.810131022$$

Solution Steps

Add 3 and 48 to get 51.

$$\frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{\sqrt{2}+\sqrt{51}}$$

Rationalize the denominator of \frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{\sqrt{2}+\sqrt{51}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{51}.

$$\frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{\left(\sqrt{2}+\sqrt{51}\right)\left(\sqrt{2}-\sqrt{51}\right)}$$

Consider \left(\sqrt{2}+\sqrt{51}\right)\left(\sqrt{2}-\sqrt{51}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{51}\right)^{2}}$$

Square \sqrt{2}. Square \sqrt{51}.

$$\frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{2-51}$$

Subtract 51 from 2 to get -49.

$$\frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\times \frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{-49}$$

Multiply \frac{6+3\sqrt{10}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} times \frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{-49} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(6+3\sqrt{10}\right)\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(\sqrt{2}-\sqrt{51}\right)}{\left(\sqrt{2}+\sqrt{3}+\sqrt{5}\right)\left(-49\right)}$$

Cancel out \sqrt{2}+\sqrt{3}+\sqrt{5} in both numerator and denominator.

$$\frac{\left(3\sqrt{10}+6\right)\left(\sqrt{2}-\sqrt{51}\right)}{-49}$$

Apply the distributive property by multiplying each term of 3\sqrt{10}+6 by each term of \sqrt{2}-\sqrt{51}.

$$\frac{3\sqrt{10}\sqrt{2}-3\sqrt{51}\sqrt{10}+6\sqrt{2}-6\sqrt{51}}{-49}$$

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}.

$$\frac{3\sqrt{2}\sqrt{5}\sqrt{2}-3\sqrt{51}\sqrt{10}+6\sqrt{2}-6\sqrt{51}}{-49}$$

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

$$\frac{3\times 2\sqrt{5}-3\sqrt{51}\sqrt{10}+6\sqrt{2}-6\sqrt{51}}{-49}$$

Multiply 3 and 2 to get 6.

$$\frac{6\sqrt{5}-3\sqrt{51}\sqrt{10}+6\sqrt{2}-6\sqrt{51}}{-49}$$

To multiply \sqrt{51} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{6\sqrt{5}-3\sqrt{510}+6\sqrt{2}-6\sqrt{51}}{-49}$$

Factor

$$\frac{3 {(\sqrt{510} + 2 \sqrt{51} - 2 \sqrt{2} - 2 \sqrt{5})}}{49} = 1.8101310219558586$$