Solve sqrt{(1-2/7)*(7/3+1/15)*(5/12)^2-(4-frac{1}{4})*(frac{7}{5}-frac{19}{15})^2})

$$\sqrt { ( 1 - \frac { 2 } { 7 } ) \times ( \frac { 7 } { 3 } + \frac { 1 } { 15 } ) \times ( \frac { 5 } { 12 } ) ^ { 2 } - ( 4 - \frac { 1 } { 4 } ) \times ( \frac { 7 } { 5 } - \frac { 19 } { 15 } ) ^ { 2 } } )$$

Evaluate

$$\frac{\sqrt{10185}}{210}\approx 0.480575052$$

Solution Steps

Subtract \frac{2}{7} from 1 to get \frac{5}{7}.

$$\sqrt{\frac{5}{7}\left(\frac{7}{3}+\frac{1}{15}\right)\times \left(\frac{5}{12}\right)^{2}-\left(4-\frac{1}{4}\right)\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Add \frac{7}{3} and \frac{1}{15} to get \frac{12}{5}.

$$\sqrt{\frac{5}{7}\times \frac{12}{5}\times \left(\frac{5}{12}\right)^{2}-\left(4-\frac{1}{4}\right)\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Multiply \frac{5}{7} and \frac{12}{5} to get \frac{12}{7}.

$$\sqrt{\frac{12}{7}\times \left(\frac{5}{12}\right)^{2}-\left(4-\frac{1}{4}\right)\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Calculate \frac{5}{12} to the power of 2 and get \frac{25}{144}.

$$\sqrt{\frac{12}{7}\times \frac{25}{144}-\left(4-\frac{1}{4}\right)\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Multiply \frac{12}{7} and \frac{25}{144} to get \frac{25}{84}.

$$\sqrt{\frac{25}{84}-\left(4-\frac{1}{4}\right)\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Subtract \frac{1}{4} from 4 to get \frac{15}{4}.

$$\sqrt{\frac{25}{84}-\frac{15}{4}\left(\frac{7}{5}-\frac{19}{15}\right)^{2}}$$

Subtract \frac{19}{15} from \frac{7}{5} to get \frac{2}{15}.

$$\sqrt{\frac{25}{84}-\frac{15}{4}\times \left(\frac{2}{15}\right)^{2}}$$

Calculate \frac{2}{15} to the power of 2 and get \frac{4}{225}.

$$\sqrt{\frac{25}{84}-\frac{15}{4}\times \frac{4}{225}}$$

Multiply \frac{15}{4} and \frac{4}{225} to get \frac{1}{15}.

$$\sqrt{\frac{25}{84}-\frac{1}{15}}$$

Subtract \frac{1}{15} from \frac{25}{84} to get \frac{97}{420}.

$$\sqrt{\frac{97}{420}}$$

Rewrite the square root of the division \sqrt{\frac{97}{420}} as the division of square roots \frac{\sqrt{97}}{\sqrt{420}}.

$$\frac{\sqrt{97}}{\sqrt{420}}$$

Factor 420=2^{2}\times 105. Rewrite the square root of the product \sqrt{2^{2}\times 105} as the product of square roots \sqrt{2^{2}}\sqrt{105}. Take the square root of 2^{2}.

$$\frac{\sqrt{97}}{2\sqrt{105}}$$

Rationalize the denominator of \frac{\sqrt{97}}{2\sqrt{105}} by multiplying numerator and denominator by \sqrt{105}.

$$\frac{\sqrt{97}\sqrt{105}}{2\left(\sqrt{105}\right)^{2}}$$

The square of \sqrt{105} is 105.

$$\frac{\sqrt{97}\sqrt{105}}{2\times 105}$$

To multiply \sqrt{97} and \sqrt{105}, multiply the numbers under the square root.

$$\frac{\sqrt{10185}}{2\times 105}$$

Multiply 2 and 105 to get 210.

$$\frac{\sqrt{10185}}{210}$$