Solve frac{sqrt{5}+sqrt{8}}{sqrt{2}}-(sqrt{3})^2

$$\frac { \sqrt { 5 } + \sqrt { 8 } } { \sqrt { 2 } } - ( \sqrt { 3 } ) ^ { 2 }$$

Evaluate

$$\frac{\sqrt{10}}{2}-1\approx 0.58113883$$

Short Solution Steps

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}.

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

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

Since \frac{\left(\sqrt{5}+2\sqrt{2}\right)\sqrt{2}}{2} and \frac{3\times 2}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(\sqrt{5}+2\sqrt{2}\right)\sqrt{2}-3\times 2.

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

Do the calculations in \sqrt{10}+4-6.

$$\frac{\sqrt{10}-2}{2}$$