Solve sqrt{1/16+sqrt{1+5/4}-0,89}

$$\sqrt { \frac { 1 } { 16 } + \sqrt { 1 + \frac { 5 } { 4 } } - 0,89 }$$

Evaluate

$$\frac{\sqrt{269}}{20}\approx 0.820060973$$

Solution Steps

Convert 1 to fraction \frac{4}{4}.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{4}{4}+\frac{5}{4}}-0,89}$$

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

$$\sqrt{\frac{1}{16}+\sqrt{\frac{4+5}{4}}-0,89}$$

Add 4 and 5 to get 9.

$$\sqrt{\frac{1}{16}+\sqrt{\frac{9}{4}}-0,89}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\sqrt{\frac{1}{16}+\frac{3}{2}-0,89}$$

Least common multiple of 16 and 2 is 16. Convert \frac{1}{16} and \frac{3}{2} to fractions with denominator 16.

$$\sqrt{\frac{1}{16}+\frac{24}{16}-0,89}$$

Since \frac{1}{16} and \frac{24}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1+24}{16}-0,89}$$

Add 1 and 24 to get 25.

$$\sqrt{\frac{25}{16}-0,89}$$

Convert decimal number 0,89 to fraction \frac{89}{100}.

$$\sqrt{\frac{25}{16}-\frac{89}{100}}$$

Least common multiple of 16 and 100 is 400. Convert \frac{25}{16} and \frac{89}{100} to fractions with denominator 400.

$$\sqrt{\frac{625}{400}-\frac{356}{400}}$$

Since \frac{625}{400} and \frac{356}{400} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{625-356}{400}}$$

Subtract 356 from 625 to get 269.

$$\sqrt{\frac{269}{400}}$$

Rewrite the square root of the division \sqrt{\frac{269}{400}} as the division of square roots \frac{\sqrt{269}}{\sqrt{400}}.

$$\frac{\sqrt{269}}{\sqrt{400}}$$

Calculate the square root of 400 and get 20.

$$\frac{\sqrt{269}}{20}$$