Solve sqrt{1/8-sqrt{20/1+1/4}:(-4/5)}-frac{7}{4}

$$\sqrt { \frac { 1 } { 8 } - \sqrt { \frac { 20 } { 1 } + \frac { 1 } { 4 } } : ( - \frac { 4 } { 5 } ) } - \frac { 7 } { 4 }$$

Evaluate

$$\frac{\sqrt{23}}{2}-\frac{7}{4}\approx 0.647915762$$

Short Solution Steps

Anything divided by one gives itself.

$$\sqrt{\frac{1}{8}-\frac{\sqrt{20+\frac{1}{4}}}{-\frac{4}{5}}}-\frac{7}{4}$$

Convert 20 to fraction \frac{80}{4}.

$$\sqrt{\frac{1}{8}-\frac{\sqrt{\frac{80}{4}+\frac{1}{4}}}{-\frac{4}{5}}}-\frac{7}{4}$$

Since \frac{80}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1}{8}-\frac{\sqrt{\frac{80+1}{4}}}{-\frac{4}{5}}}-\frac{7}{4}$$

Add 80 and 1 to get 81.

$$\sqrt{\frac{1}{8}-\frac{\sqrt{\frac{81}{4}}}{-\frac{4}{5}}}-\frac{7}{4}$$

Rewrite the square root of the division \frac{81}{4} as the division of square roots \frac{\sqrt{81}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\sqrt{\frac{1}{8}-\frac{\frac{9}{2}}{-\frac{4}{5}}}-\frac{7}{4}$$

Divide \frac{9}{2} by -\frac{4}{5} by multiplying \frac{9}{2} by the reciprocal of -\frac{4}{5}.

$$\sqrt{\frac{1}{8}-\frac{9}{2}\left(-\frac{5}{4}\right)}-\frac{7}{4}$$

Multiply \frac{9}{2} times -\frac{5}{4} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{1}{8}-\frac{9\left(-5\right)}{2\times 4}}-\frac{7}{4}$$

Do the multiplications in the fraction \frac{9\left(-5\right)}{2\times 4}.

$$\sqrt{\frac{1}{8}-\frac{-45}{8}}-\frac{7}{4}$$

Fraction \frac{-45}{8} can be rewritten as -\frac{45}{8} by extracting the negative sign.

$$\sqrt{\frac{1}{8}-\left(-\frac{45}{8}\right)}-\frac{7}{4}$$

The opposite of -\frac{45}{8} is \frac{45}{8}.

$$\sqrt{\frac{1}{8}+\frac{45}{8}}-\frac{7}{4}$$

Since \frac{1}{8} and \frac{45}{8} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1+45}{8}}-\frac{7}{4}$$

Add 1 and 45 to get 46.

$$\sqrt{\frac{46}{8}}-\frac{7}{4}$$

Reduce the fraction \frac{46}{8} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{23}{4}}-\frac{7}{4}$$

Rewrite the square root of the division \sqrt{\frac{23}{4}} as the division of square roots \frac{\sqrt{23}}{\sqrt{4}}.

$$\frac{\sqrt{23}}{\sqrt{4}}-\frac{7}{4}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{23}}{2}-\frac{7}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{\sqrt{23}}{2} times \frac{2}{2}.

$$\frac{2\sqrt{23}}{4}-\frac{7}{4}$$

Since \frac{2\sqrt{23}}{4} and \frac{7}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\sqrt{23}-7}{4}$$

Factor

$$\frac{2 \sqrt{23} - 7}{4} = 0.6479157616563596$$