Solve sqrt{{left(2+frac{1/4-frac{3/2right)}^3{left(3/2-2/3right)}^3}{{left(frac{1}{3}+frac{1}{2}-frac{3}{4}right)}^2}}{{left(frac{3}{2}+frac{1}{3}-frac{7}{4}right)}^2}}

$$\sqrt{ \frac{ \frac{ { \left(2+ \frac{ 1 }{ 4 } - \frac{ 3 }{ 2 } \right) }^{ 3 } { \left( \frac{ 3 }{ 2 } - \frac{ 2 }{ 3 } \right) }^{ 3 } }{ { \left( \frac{ 1 }{ 3 } + \frac{ 1 }{ 2 } - \frac{ 3 }{ 4 } \right) }^{ 2 } } }{ { \left( \frac{ 3 }{ 2 } + \frac{ 1 }{ 3 } - \frac{ 7 }{ 4 } \right) }^{ 2 } } }$$

Evaluate

$$\frac{45\sqrt{10}}{2}\approx 71.151247354$$

Solution Steps

Add 2 and \frac{1}{4} to get \frac{9}{4}.

$$\sqrt{\frac{\frac{\left(\frac{9}{4}-\frac{3}{2}\right)^{3}\left(\frac{3}{2}-\frac{2}{3}\right)^{3}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Subtract \frac{3}{2} from \frac{9}{4} to get \frac{3}{4}.

$$\sqrt{\frac{\frac{\left(\frac{3}{4}\right)^{3}\left(\frac{3}{2}-\frac{2}{3}\right)^{3}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Calculate \frac{3}{4} to the power of 3 and get \frac{27}{64}.

$$\sqrt{\frac{\frac{\frac{27}{64}\left(\frac{3}{2}-\frac{2}{3}\right)^{3}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Subtract \frac{2}{3} from \frac{3}{2} to get \frac{5}{6}.

$$\sqrt{\frac{\frac{\frac{27}{64}\times \left(\frac{5}{6}\right)^{3}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Calculate \frac{5}{6} to the power of 3 and get \frac{125}{216}.

$$\sqrt{\frac{\frac{\frac{27}{64}\times \frac{125}{216}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Multiply \frac{27}{64} and \frac{125}{216} to get \frac{125}{512}.

$$\sqrt{\frac{\frac{\frac{125}{512}}{\left(\frac{1}{3}+\frac{1}{2}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Add \frac{1}{3} and \frac{1}{2} to get \frac{5}{6}.

$$\sqrt{\frac{\frac{\frac{125}{512}}{\left(\frac{5}{6}-\frac{3}{4}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Subtract \frac{3}{4} from \frac{5}{6} to get \frac{1}{12}.

$$\sqrt{\frac{\frac{\frac{125}{512}}{\left(\frac{1}{12}\right)^{2}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Calculate \frac{1}{12} to the power of 2 and get \frac{1}{144}.

$$\sqrt{\frac{\frac{\frac{125}{512}}{\frac{1}{144}}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Divide \frac{125}{512} by \frac{1}{144} by multiplying \frac{125}{512} by the reciprocal of \frac{1}{144}.

$$\sqrt{\frac{\frac{125}{512}\times 144}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Multiply \frac{125}{512} and 144 to get \frac{1125}{32}.

$$\sqrt{\frac{\frac{1125}{32}}{\left(\frac{3}{2}+\frac{1}{3}-\frac{7}{4}\right)^{2}}}$$

Add \frac{3}{2} and \frac{1}{3} to get \frac{11}{6}.

$$\sqrt{\frac{\frac{1125}{32}}{\left(\frac{11}{6}-\frac{7}{4}\right)^{2}}}$$

Subtract \frac{7}{4} from \frac{11}{6} to get \frac{1}{12}.

$$\sqrt{\frac{\frac{1125}{32}}{\left(\frac{1}{12}\right)^{2}}}$$

Calculate \frac{1}{12} to the power of 2 and get \frac{1}{144}.

$$\sqrt{\frac{\frac{1125}{32}}{\frac{1}{144}}}$$

Divide \frac{1125}{32} by \frac{1}{144} by multiplying \frac{1125}{32} by the reciprocal of \frac{1}{144}.

$$\sqrt{\frac{1125}{32}\times 144}$$

Multiply \frac{1125}{32} and 144 to get \frac{10125}{2}.

$$\sqrt{\frac{10125}{2}}$$

Rewrite the square root of the division \sqrt{\frac{10125}{2}} as the division of square roots \frac{\sqrt{10125}}{\sqrt{2}}.

$$\frac{\sqrt{10125}}{\sqrt{2}}$$

Factor 10125=45^{2}\times 5. Rewrite the square root of the product \sqrt{45^{2}\times 5} as the product of square roots \sqrt{45^{2}}\sqrt{5}. Take the square root of 45^{2}.

$$\frac{45\sqrt{5}}{\sqrt{2}}$$

Rationalize the denominator of \frac{45\sqrt{5}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{45\sqrt{5}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{45\sqrt{5}\sqrt{2}}{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{45\sqrt{10}}{2}$$