Solve frac{frac{left(3^7.3right)^-42^2/3^4{2^{-2}-{4}^{2}}}{3/4{left(frac{2/3right)}^{2}2/6}{frac{{3}^{2}}{4}frac{{2}^{3}}{2}frac{{2}^{3}}{3}}}}{frac{frac{{3}^{4}-27-64}{{2}^{4}2{2}^{3}}}{frac{4*16*27{3}^{-2}}{{2}^6{3}^-2{3}^3{2}^6}}}

$$\frac{ \frac{ \frac{ { \left( { 3 }^{ 7 } .3 \right) }^{ -4 } { 2 }^{ 2 } }{ { 3 }^{ 4 } { 2 }^{ -2 } - { 4 }^{ 2 } } }{ \frac{ \frac{ 3 }{ 4 } { \left( \frac{ 2 }{ 3 } \right) }^{ 2 } \frac{ 2 }{ 6 } }{ \frac{ { 3 }^{ 2 } }{ 4 } \frac{ { 2 }^{ 3 } }{ 2 } \frac{ { 2 }^{ 3 } }{ 3 } } } }{ \frac{ \frac{ { 3 }^{ 4 } -27-64 }{ { 2 }^{ 4 } 2 { 2 }^{ 3 } } }{ \frac{ 4 \cdot 16 \cdot 27 { 3 }^{ -2 } }{ { 2 }^{ 6 } { 3 }^{ -2 } { 3 }^{ 3 } { 2 }^{ 6 } } } }$$

Evaluate

$$-\frac{512000}{1166716415203011}\approx -4.388384301 \cdot 10^{-10}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 3 to get 2.

$$\frac{\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}}}{\frac{\frac{3^{4}-27-64}{2^{4}\times 2\times 2^{3}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{6}\times 3^{-2}\times 3^{3}\times 2^{6}}}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}}}{\frac{\frac{3^{4}-27-64}{2^{5}\times 2^{3}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{6}\times 3^{-2}\times 3^{3}\times 2^{6}}}}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}}}{\frac{\frac{3^{4}-27-64}{2^{8}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{6}\times 3^{-2}\times 3^{3}\times 2^{6}}}}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}}}{\frac{\frac{3^{4}-27-64}{2^{8}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{-2}\times 3^{3}}}}$$

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$\frac{\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}}}{\frac{\frac{3^{4}-27-64}{2^{8}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}}$$

Divide \frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}} by \frac{\frac{3^{4}-27-64}{2^{8}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}} by multiplying \frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}} by the reciprocal of \frac{\frac{3^{4}-27-64}{2^{8}}}{\frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}.

$$\frac{\frac{\left(3^{7}\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 3 to the power of 7 and get 2187.

$$\frac{\frac{\left(2187\times 0.3\right)^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 2187 and 0.3 to get 656.1.

$$\frac{\frac{656.1^{-4}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 656.1 to the power of -4 and get \frac{10000}{1853020188851841}.

$$\frac{\frac{\frac{10000}{1853020188851841}\times 2^{2}}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{\frac{10000}{1853020188851841}\times 4}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{10000}{1853020188851841} and 4 to get \frac{40000}{1853020188851841}.

$$\frac{\frac{\frac{40000}{1853020188851841}}{3^{4}\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{\frac{\frac{40000}{1853020188851841}}{81\times 2^{-2}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{\frac{40000}{1853020188851841}}{81\times \frac{1}{4}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 81 and \frac{1}{4} to get \frac{81}{4}.

$$\frac{\frac{\frac{40000}{1853020188851841}}{\frac{81}{4}-4^{2}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\frac{\frac{40000}{1853020188851841}}{\frac{81}{4}-16}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Subtract 16 from \frac{81}{4} to get \frac{17}{4}.

$$\frac{\frac{\frac{40000}{1853020188851841}}{\frac{17}{4}}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Divide \frac{40000}{1853020188851841} by \frac{17}{4} by multiplying \frac{40000}{1853020188851841} by the reciprocal of \frac{17}{4}.

$$\frac{\frac{40000}{1853020188851841}\times \frac{4}{17}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{40000}{1853020188851841} and \frac{4}{17} to get \frac{160000}{31501343210481297}.

$$\frac{\frac{160000}{31501343210481297}\times \frac{4\times 16\times 27\times 3^{-2}}{2^{12}\times 3^{1}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{160000}{31501343210481297}\times \frac{4\times 9\times 16\times 3^{-2}}{2^{12}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 4 and 9 to get 36.

$$\frac{\frac{160000}{31501343210481297}\times \frac{36\times 16\times 3^{-2}}{2^{12}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 36 and 16 to get 576.

$$\frac{\frac{160000}{31501343210481297}\times \frac{576\times 3^{-2}}{2^{12}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\frac{160000}{31501343210481297}\times \frac{576\times \frac{1}{9}}{2^{12}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 576 and \frac{1}{9} to get 64.

$$\frac{\frac{160000}{31501343210481297}\times \frac{64}{2^{12}}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 2 to the power of 12 and get 4096.

$$\frac{\frac{160000}{31501343210481297}\times \frac{64}{4096}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Reduce the fraction \frac{64}{4096} to lowest terms by extracting and canceling out 64.

$$\frac{\frac{160000}{31501343210481297}\times \frac{1}{64}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{160000}{31501343210481297} and \frac{1}{64} to get \frac{2500}{31501343210481297}.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{3}{4}\times \left(\frac{2}{3}\right)^{2}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{3}{4}\times \frac{4}{9}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{3}{4} and \frac{4}{9} to get \frac{1}{3}.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{3}\times \frac{2}{6}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

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

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{3}\times \frac{1}{3}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{1}{3} and \frac{1}{3} to get \frac{1}{9}.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{\frac{3^{2}}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{\frac{9}{4}\times 2^{2}\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{\frac{9}{4}\times 4\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply \frac{9}{4} and 4 to get 9.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{9\times \frac{2^{3}}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{9\times \frac{8}{3}}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 9 and \frac{8}{3} to get 24.

$$\frac{\frac{2500}{31501343210481297}}{\frac{\frac{1}{9}}{24}\times \frac{3^{4}-27-64}{2^{8}}}$$

Express \frac{\frac{1}{9}}{24} as a single fraction.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{9\times 24}\times \frac{3^{4}-27-64}{2^{8}}}$$

Multiply 9 and 24 to get 216.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\times \frac{3^{4}-27-64}{2^{8}}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\times \frac{81-27-64}{2^{8}}}$$

Subtract 27 from 81 to get 54.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\times \frac{54-64}{2^{8}}}$$

Subtract 64 from 54 to get -10.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\times \frac{-10}{2^{8}}}$$

Calculate 2 to the power of 8 and get 256.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\times \frac{-10}{256}}$$

Reduce the fraction \frac{-10}{256} to lowest terms by extracting and canceling out 2.

$$\frac{\frac{2500}{31501343210481297}}{\frac{1}{216}\left(-\frac{5}{128}\right)}$$

Multiply \frac{1}{216} and -\frac{5}{128} to get -\frac{5}{27648}.

$$\frac{\frac{2500}{31501343210481297}}{-\frac{5}{27648}}$$

Divide \frac{2500}{31501343210481297} by -\frac{5}{27648} by multiplying \frac{2500}{31501343210481297} by the reciprocal of -\frac{5}{27648}.

$$\frac{2500}{31501343210481297}\left(-\frac{27648}{5}\right)$$

Multiply \frac{2500}{31501343210481297} and -\frac{27648}{5} to get -\frac{512000}{1166716415203011}.

$$-\frac{512000}{1166716415203011}$$

Factor

$$-\frac{512000}{1166716415203011} = -4.3883843008321 \times 10^{-10}$$