Solve {left(frac{2/3right)}^4{left(3/2right)}^5+2left(-1/4+sqrt{6+-2^2+3*5/64}right)-frac{1}{2^2}}{1+frac{frac{2}{7}-frac{1}{5}}{frac{1}{4}-frac{2}{5}}-left(1-frac{2}{5}right)left(frac{2}{3}-left(frac{3}{4}-frac{2}{5}right)left(1+frac{3}{7}right)right)}

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

Evaluate

$$\frac{140}{23}\approx 6.086956522$$

Solution Steps

Calculate \frac{2}{3} to the power of 4 and get \frac{16}{81}.

$$\frac{\frac{16}{81}\times \left(\frac{3}{2}\right)^{5}+2\left(-\frac{1}{4}+\sqrt{\frac{6+\left(-2\right)^{2}+3\times 5}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Calculate \frac{3}{2} to the power of 5 and get \frac{243}{32}.

$$\frac{\frac{16}{81}\times \frac{243}{32}+2\left(-\frac{1}{4}+\sqrt{\frac{6+\left(-2\right)^{2}+3\times 5}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Multiply \frac{16}{81} times \frac{243}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{16\times 243}{81\times 32}+2\left(-\frac{1}{4}+\sqrt{\frac{6+\left(-2\right)^{2}+3\times 5}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Do the multiplications in the fraction \frac{16\times 243}{81\times 32}.

$$\frac{\frac{3888}{2592}+2\left(-\frac{1}{4}+\sqrt{\frac{6+\left(-2\right)^{2}+3\times 5}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Reduce the fraction \frac{3888}{2592} to lowest terms by extracting and canceling out 1296.

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

Calculate -2 to the power of 2 and get 4.

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

Add 6 and 4 to get 10.

$$\frac{\frac{3}{2}+2\left(-\frac{1}{4}+\sqrt{\frac{10+3\times 5}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{3}{2}+2\left(-\frac{1}{4}+\sqrt{\frac{10+15}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Add 10 and 15 to get 25.

$$\frac{\frac{3}{2}+2\left(-\frac{1}{4}+\sqrt{\frac{25}{64}}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Rewrite the square root of the division \frac{25}{64} as the division of square roots \frac{\sqrt{25}}{\sqrt{64}}. Take the square root of both numerator and denominator.

$$\frac{\frac{3}{2}+2\left(-\frac{1}{4}+\frac{5}{8}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Least common multiple of 4 and 8 is 8. Convert -\frac{1}{4} and \frac{5}{8} to fractions with denominator 8.

$$\frac{\frac{3}{2}+2\left(-\frac{2}{8}+\frac{5}{8}\right)-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since -\frac{2}{8} and \frac{5}{8} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3}{2}+2\times \frac{-2+5}{8}-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Add -2 and 5 to get 3.

$$\frac{\frac{3}{2}+2\times \frac{3}{8}-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Express 2\times \frac{3}{8} as a single fraction.

$$\frac{\frac{3}{2}+\frac{2\times 3}{8}-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{3}{2}+\frac{6}{8}-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

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

Least common multiple of 2 and 4 is 4. Convert \frac{3}{2} and \frac{3}{4} to fractions with denominator 4.

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

Since \frac{6}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

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

Add 6 and 3 to get 9.

$$\frac{\frac{9}{4}-\frac{1}{2^{2}}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{9}{4}-\frac{1}{4}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since \frac{9}{4} and \frac{1}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{9-1}{4}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Subtract 1 from 9 to get 8.

$$\frac{\frac{8}{4}}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Divide 8 by 4 to get 2.

$$\frac{2}{1+\frac{\frac{2}{7}-\frac{1}{5}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Least common multiple of 7 and 5 is 35. Convert \frac{2}{7} and \frac{1}{5} to fractions with denominator 35.

$$\frac{2}{1+\frac{\frac{10}{35}-\frac{7}{35}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since \frac{10}{35} and \frac{7}{35} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{1+\frac{\frac{10-7}{35}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Subtract 7 from 10 to get 3.

$$\frac{2}{1+\frac{\frac{3}{35}}{\frac{1}{4}-\frac{2}{5}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Least common multiple of 4 and 5 is 20. Convert \frac{1}{4} and \frac{2}{5} to fractions with denominator 20.

$$\frac{2}{1+\frac{\frac{3}{35}}{\frac{5}{20}-\frac{8}{20}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since \frac{5}{20} and \frac{8}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{1+\frac{\frac{3}{35}}{\frac{5-8}{20}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Subtract 8 from 5 to get -3.

$$\frac{2}{1+\frac{\frac{3}{35}}{-\frac{3}{20}}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Divide \frac{3}{35} by -\frac{3}{20} by multiplying \frac{3}{35} by the reciprocal of -\frac{3}{20}.

$$\frac{2}{1+\frac{3}{35}\left(-\frac{20}{3}\right)-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Multiply \frac{3}{35} times -\frac{20}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{1+\frac{3\left(-20\right)}{35\times 3}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2}{1+\frac{-20}{35}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Reduce the fraction \frac{-20}{35} to lowest terms by extracting and canceling out 5.

$$\frac{2}{1-\frac{4}{7}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Convert 1 to fraction \frac{7}{7}.

$$\frac{2}{\frac{7}{7}-\frac{4}{7}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

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

$$\frac{2}{\frac{7-4}{7}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Subtract 4 from 7 to get 3.

$$\frac{2}{\frac{3}{7}-\left(1-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Convert 1 to fraction \frac{5}{5}.

$$\frac{2}{\frac{3}{7}-\left(\frac{5}{5}-\frac{2}{5}\right)\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since \frac{5}{5} and \frac{2}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{\frac{3}{7}-\frac{5-2}{5}\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Subtract 2 from 5 to get 3.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\left(\frac{3}{4}-\frac{2}{5}\right)\left(1+\frac{3}{7}\right)\right)}$$

Least common multiple of 4 and 5 is 20. Convert \frac{3}{4} and \frac{2}{5} to fractions with denominator 20.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\left(\frac{15}{20}-\frac{8}{20}\right)\left(1+\frac{3}{7}\right)\right)}$$

Since \frac{15}{20} and \frac{8}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{15-8}{20}\left(1+\frac{3}{7}\right)\right)}$$

Subtract 8 from 15 to get 7.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{7}{20}\left(1+\frac{3}{7}\right)\right)}$$

Convert 1 to fraction \frac{7}{7}.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{7}{20}\left(\frac{7}{7}+\frac{3}{7}\right)\right)}$$

Since \frac{7}{7} and \frac{3}{7} have the same denominator, add them by adding their numerators.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{7}{20}\times \frac{7+3}{7}\right)}$$

Add 7 and 3 to get 10.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{7}{20}\times \frac{10}{7}\right)}$$

Multiply \frac{7}{20} times \frac{10}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{7\times 10}{20\times 7}\right)}$$

Cancel out 7 in both numerator and denominator.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{10}{20}\right)}$$

Reduce the fraction \frac{10}{20} to lowest terms by extracting and canceling out 10.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{2}{3}-\frac{1}{2}\right)}$$

Least common multiple of 3 and 2 is 6. Convert \frac{2}{3} and \frac{1}{2} to fractions with denominator 6.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\left(\frac{4}{6}-\frac{3}{6}\right)}$$

Since \frac{4}{6} and \frac{3}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\times \frac{4-3}{6}}$$

Subtract 3 from 4 to get 1.

$$\frac{2}{\frac{3}{7}-\frac{3}{5}\times \frac{1}{6}}$$

Multiply \frac{3}{5} times \frac{1}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{\frac{3}{7}-\frac{3\times 1}{5\times 6}}$$

Do the multiplications in the fraction \frac{3\times 1}{5\times 6}.

$$\frac{2}{\frac{3}{7}-\frac{3}{30}}$$

Reduce the fraction \frac{3}{30} to lowest terms by extracting and canceling out 3.

$$\frac{2}{\frac{3}{7}-\frac{1}{10}}$$

Least common multiple of 7 and 10 is 70. Convert \frac{3}{7} and \frac{1}{10} to fractions with denominator 70.

$$\frac{2}{\frac{30}{70}-\frac{7}{70}}$$

Since \frac{30}{70} and \frac{7}{70} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2}{\frac{30-7}{70}}$$

Subtract 7 from 30 to get 23.

$$\frac{2}{\frac{23}{70}}$$

Divide 2 by \frac{23}{70} by multiplying 2 by the reciprocal of \frac{23}{70}.

$$2\times \frac{70}{23}$$

Express 2\times \frac{70}{23} as a single fraction.

$$\frac{2\times 70}{23}$$

Multiply 2 and 70 to get 140.

$$\frac{140}{23}$$

Factor

$$\frac{2 ^ {2} \cdot 5 \cdot 7}{23} = 6\frac{2}{23} = 6.086956521739131$$