Solve frac{2sqrt{5}4+4sqrt{6}}{4sqrt{8}-3sqrt{2}}

$$\frac { 2 \sqrt { 5 } 4 + 4 \sqrt { 6 } } { 4 \sqrt { 8 } - 3 \sqrt { 2 } }$$

Evaluate

$$\frac{4\left(\sqrt{3}+\sqrt{10}\right)}{5}\approx 3.915462774$$

Solution Steps

Multiply 2 and 4 to get 8.

$$\frac{8\sqrt{5}+4\sqrt{6}}{4\sqrt{8}-3\sqrt{2}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{8\sqrt{5}+4\sqrt{6}}{4\times 2\sqrt{2}-3\sqrt{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{8\sqrt{5}+4\sqrt{6}}{8\sqrt{2}-3\sqrt{2}}$$

Combine 8\sqrt{2} and -3\sqrt{2} to get 5\sqrt{2}.

$$\frac{8\sqrt{5}+4\sqrt{6}}{5\sqrt{2}}$$

Rationalize the denominator of \frac{8\sqrt{5}+4\sqrt{6}}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(8\sqrt{5}+4\sqrt{6}\right)\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(8\sqrt{5}+4\sqrt{6}\right)\sqrt{2}}{5\times 2}$$

Multiply 5 and 2 to get 10.

$$\frac{\left(8\sqrt{5}+4\sqrt{6}\right)\sqrt{2}}{10}$$

Use the distributive property to multiply 8\sqrt{5}+4\sqrt{6} by \sqrt{2}.

$$\frac{8\sqrt{5}\sqrt{2}+4\sqrt{6}\sqrt{2}}{10}$$

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

$$\frac{8\sqrt{10}+4\sqrt{6}\sqrt{2}}{10}$$

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}.

$$\frac{8\sqrt{10}+4\sqrt{2}\sqrt{3}\sqrt{2}}{10}$$

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

$$\frac{8\sqrt{10}+4\times 2\sqrt{3}}{10}$$

Multiply 4 and 2 to get 8.

$$\frac{8\sqrt{10}+8\sqrt{3}}{10}$$

Factor

$$\frac{4 {(\sqrt{3} + \sqrt{10})}}{5} = 3.9154627741898054$$