Solve sqrt{1-1/2+frac{1/4}-(1+1/8+1/16)}{frac{1}{4}}

$$\frac { \sqrt { 1 - \frac { 1 } { 2 } + \frac { 1 } { 4 } } - ( 1 + \frac { 1 } { 8 } + \frac { 1 } { 16 } ) } { \frac { 1 } { 4 } }$$

Evaluate

$$2\sqrt{3}-\frac{19}{4}\approx -1.285898385$$

Short Solution Steps

Convert 1 to fraction \frac{2}{2}.

$$\frac{\sqrt{\frac{2}{2}-\frac{1}{2}+\frac{1}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{\frac{2-1}{2}+\frac{1}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Subtract 1 from 2 to get 1.

$$\frac{\sqrt{\frac{1}{2}+\frac{1}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Least common multiple of 2 and 4 is 4. Convert \frac{1}{2} and \frac{1}{4} to fractions with denominator 4.

$$\frac{\sqrt{\frac{2}{4}+\frac{1}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Since \frac{2}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{\frac{2+1}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Add 2 and 1 to get 3.

$$\frac{\sqrt{\frac{3}{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$\frac{\frac{\sqrt{3}}{\sqrt{4}}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{\sqrt{3}}{2}-\left(1+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Convert 1 to fraction \frac{8}{8}.

$$\frac{\frac{\sqrt{3}}{2}-\left(\frac{8}{8}+\frac{1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Since \frac{8}{8} and \frac{1}{8} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\sqrt{3}}{2}-\left(\frac{8+1}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Add 8 and 1 to get 9.

$$\frac{\frac{\sqrt{3}}{2}-\left(\frac{9}{8}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Least common multiple of 8 and 16 is 16. Convert \frac{9}{8} and \frac{1}{16} to fractions with denominator 16.

$$\frac{\frac{\sqrt{3}}{2}-\left(\frac{18}{16}+\frac{1}{16}\right)}{\frac{1}{4}}$$

Since \frac{18}{16} and \frac{1}{16} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\sqrt{3}}{2}-\frac{18+1}{16}}{\frac{1}{4}}$$

Add 18 and 1 to get 19.

$$\frac{\frac{\sqrt{3}}{2}-\frac{19}{16}}{\frac{1}{4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 16 is 16. Multiply \frac{\sqrt{3}}{2} times \frac{8}{8}.

$$\frac{\frac{8\sqrt{3}}{16}-\frac{19}{16}}{\frac{1}{4}}$$

Since \frac{8\sqrt{3}}{16} and \frac{19}{16} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{8\sqrt{3}-19}{16}}{\frac{1}{4}}$$

Divide \frac{8\sqrt{3}-19}{16} by \frac{1}{4} by multiplying \frac{8\sqrt{3}-19}{16} by the reciprocal of \frac{1}{4}.

$$\frac{\left(8\sqrt{3}-19\right)\times 4}{16}$$

Divide \left(8\sqrt{3}-19\right)\times 4 by 16 to get \left(8\sqrt{3}-19\right)\times \frac{1}{4}.

$$\left(8\sqrt{3}-19\right)\times \frac{1}{4}$$

Use the distributive property to multiply 8\sqrt{3}-19 by \frac{1}{4}.

$$8\sqrt{3}\times \frac{1}{4}-19\times \frac{1}{4}$$

Multiply 8 and \frac{1}{4} to get \frac{8}{4}.

$$\frac{8}{4}\sqrt{3}-19\times \frac{1}{4}$$

Divide 8 by 4 to get 2.

$$2\sqrt{3}-19\times \frac{1}{4}$$

Multiply -19 and \frac{1}{4} to get \frac{-19}{4}.

$$2\sqrt{3}+\frac{-19}{4}$$

Fraction \frac{-19}{4} can be rewritten as -\frac{19}{4} by extracting the negative sign.

$$2\sqrt{3}-\frac{19}{4}$$

Factor

$$\frac{8 \sqrt{3} - 19}{4} = -1.2858983848622456$$