Solve frac{frac{sqrt{15}*frac{sqrt{2}}{2}}{sqrt{6}+sqrt{2}}}{4}

$$\frac { \frac { \sqrt { 15 } \times \frac { \sqrt { 2 } } { 2 } } { \sqrt { 6 } + \sqrt { 2 } } } { 4 }$$

Evaluate

$$\frac{3\sqrt{5}-\sqrt{15}}{16}\approx 0.177201287$$

Solution Steps

Express \frac{\frac{\sqrt{15}\times \frac{\sqrt{2}}{2}}{\sqrt{6}+\sqrt{2}}}{4} as a single fraction.

$$\frac{\sqrt{15}\times \frac{\sqrt{2}}{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 4}$$

Express \sqrt{15}\times \frac{\sqrt{2}}{2} as a single fraction.

$$\frac{\frac{\sqrt{15}\sqrt{2}}{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 4}$$

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

$$\frac{\frac{\sqrt{30}}{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 4}$$

Express \frac{\frac{\sqrt{30}}{2}}{\left(\sqrt{6}+\sqrt{2}\right)\times 4} as a single fraction.

$$\frac{\sqrt{30}}{2\left(\sqrt{6}+\sqrt{2}\right)\times 4}$$

Multiply 2 and 4 to get 8.

$$\frac{\sqrt{30}}{8\left(\sqrt{6}+\sqrt{2}\right)}$$

Use the distributive property to multiply 8 by \sqrt{6}+\sqrt{2}.

$$\frac{\sqrt{30}}{8\sqrt{6}+8\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{30}}{8\sqrt{6}+8\sqrt{2}} by multiplying numerator and denominator by 8\sqrt{6}-8\sqrt{2}.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{\left(8\sqrt{6}+8\sqrt{2}\right)\left(8\sqrt{6}-8\sqrt{2}\right)}$$

Consider \left(8\sqrt{6}+8\sqrt{2}\right)\left(8\sqrt{6}-8\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}.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{\left(8\sqrt{6}\right)^{2}-\left(8\sqrt{2}\right)^{2}}$$

Expand \left(8\sqrt{6}\right)^{2}.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{8^{2}\left(\sqrt{6}\right)^{2}-\left(8\sqrt{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{64\left(\sqrt{6}\right)^{2}-\left(8\sqrt{2}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{64\times 6-\left(8\sqrt{2}\right)^{2}}$$

Multiply 64 and 6 to get 384.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{384-\left(8\sqrt{2}\right)^{2}}$$

Expand \left(8\sqrt{2}\right)^{2}.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{384-8^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{384-64\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{384-64\times 2}$$

Multiply 64 and 2 to get 128.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{384-128}$$

Subtract 128 from 384 to get 256.

$$\frac{\sqrt{30}\left(8\sqrt{6}-8\sqrt{2}\right)}{256}$$

Use the distributive property to multiply \sqrt{30} by 8\sqrt{6}-8\sqrt{2}.

$$\frac{8\sqrt{30}\sqrt{6}-8\sqrt{30}\sqrt{2}}{256}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$\frac{8\sqrt{6}\sqrt{5}\sqrt{6}-8\sqrt{30}\sqrt{2}}{256}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{8\times 6\sqrt{5}-8\sqrt{30}\sqrt{2}}{256}$$

Multiply 8 and 6 to get 48.

$$\frac{48\sqrt{5}-8\sqrt{30}\sqrt{2}}{256}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

$$\frac{48\sqrt{5}-8\sqrt{2}\sqrt{15}\sqrt{2}}{256}$$

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

$$\frac{48\sqrt{5}-8\times 2\sqrt{15}}{256}$$

Multiply -8 and 2 to get -16.

$$\frac{48\sqrt{5}-16\sqrt{15}}{256}$$