Solve sqrt{1+4/1+frac{1{7}}}

$$\sqrt { \frac { 1 + 4 } { 1 + \frac { 1 } { 7 } } }$$

Evaluate

$$\frac{\sqrt{70}}{4}\approx 2.091650066$$

Solution Steps

Add 1 and 4 to get 5.

$$\sqrt{\frac{5}{1+\frac{1}{7}}}$$

Convert 1 to fraction \frac{7}{7}.

$$\sqrt{\frac{5}{\frac{7}{7}+\frac{1}{7}}}$$

Since \frac{7}{7} and \frac{1}{7} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5}{\frac{7+1}{7}}}$$

Add 7 and 1 to get 8.

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

Divide 5 by \frac{8}{7} by multiplying 5 by the reciprocal of \frac{8}{7}.

$$\sqrt{5\times \frac{7}{8}}$$

Express 5\times \frac{7}{8} as a single fraction.

$$\sqrt{\frac{5\times 7}{8}}$$

Multiply 5 and 7 to get 35.

$$\sqrt{\frac{35}{8}}$$

Rewrite the square root of the division \sqrt{\frac{35}{8}} as the division of square roots \frac{\sqrt{35}}{\sqrt{8}}.

$$\frac{\sqrt{35}}{\sqrt{8}}$$

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}.

$$\frac{\sqrt{35}}{2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{35}}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{35}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{35}\sqrt{2}}{2\times 2}$$

To multiply \sqrt{35} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{70}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{70}}{4}$$