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

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

Evaluate

$$\frac{\sqrt{6}}{2}-\frac{5\sqrt{10}}{2}+4\approx -2.680949279$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}+4\sqrt{2}-5\sqrt{5}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(\sqrt{3}+4\sqrt{2}-5\sqrt{5}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{3}+4\sqrt{2}-5\sqrt{5}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply \sqrt{3}+4\sqrt{2}-5\sqrt{5} by \sqrt{2}.

$$\frac{\sqrt{3}\sqrt{2}+4\left(\sqrt{2}\right)^{2}-5\sqrt{5}\sqrt{2}}{2}$$

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

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

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

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