Solve 1/sqrt{5+2}+1/3sqrt{8}

$$\frac{ 1 }{ \sqrt{ 5+2 } } + \frac{ 1 }{ 3 \sqrt{ 8 } }$$

Evaluate

$$\frac{\sqrt{7}\left(\sqrt{14}+12\right)}{84}\approx 0.495815603$$

Short Solution Steps

Add 5 and 2 to get 7.

$$\frac{1}{\sqrt{7}}+\frac{1}{3\sqrt{8}}$$

Rationalize the denominator of \frac{1}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{7}}{\left(\sqrt{7}\right)^{2}}+\frac{1}{3\sqrt{8}}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{7}}{7}+\frac{1}{3\sqrt{8}}$$

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}.

$$\frac{\sqrt{7}}{7}+\frac{1}{3\times 2\sqrt{2}}$$

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{7}}{7}+\frac{1}{6\sqrt{2}}$$

Rationalize the denominator of \frac{1}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{7}}{7}+\frac{\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{7}}{7}+\frac{\sqrt{2}}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$\frac{\sqrt{7}}{7}+\frac{\sqrt{2}}{12}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 12 is 84. Multiply \frac{\sqrt{7}}{7} times \frac{12}{12}. Multiply \frac{\sqrt{2}}{12} times \frac{7}{7}.

$$\frac{12\sqrt{7}}{84}+\frac{7\sqrt{2}}{84}$$

Since \frac{12\sqrt{7}}{84} and \frac{7\sqrt{2}}{84} have the same denominator, add them by adding their numerators.

$$\frac{12\sqrt{7}+7\sqrt{2}}{84}$$

Factor

$$\frac{\sqrt{7} {(\sqrt{2} \sqrt{7} + 12)}}{84} = 0.49581560320698514$$