Solve sqrt{25/8}+3sqrt{7}

$$\sqrt{ \frac{ 25 }{ 8 } } +3 \sqrt{ 7 }$$

Evaluate

$$\frac{5\sqrt{2}}{4}+3\sqrt{7}\approx 9.705020886$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{25}{8}} as the division of square roots \frac{\sqrt{25}}{\sqrt{8}}.

$$\frac{\sqrt{25}}{\sqrt{8}}+3\sqrt{7}$$

Calculate the square root of 25 and get 5.

$$\frac{5}{\sqrt{8}}+3\sqrt{7}$$

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}.

$$\frac{5}{2\sqrt{2}}+3\sqrt{7}$$

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

$$\frac{5\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+3\sqrt{7}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}}{2\times 2}+3\sqrt{7}$$

Multiply 2 and 2 to get 4.

$$\frac{5\sqrt{2}}{4}+3\sqrt{7}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{7} times \frac{4}{4}.

$$\frac{5\sqrt{2}}{4}+\frac{4\times 3\sqrt{7}}{4}$$

Since \frac{5\sqrt{2}}{4} and \frac{4\times 3\sqrt{7}}{4} have the same denominator, add them by adding their numerators.

$$\frac{5\sqrt{2}+4\times 3\sqrt{7}}{4}$$

Do the multiplications in 5\sqrt{2}+4\times 3\sqrt{7}.

$$\frac{5\sqrt{2}+12\sqrt{7}}{4}$$