Solve frac{(2^{3}*5^{2})^{6}*(3^{2}*2^{5})^{4}}{(2^{3}*3*5^2)^5*(2^735)^3}

$$\frac { ( 2 ^ { 3 } \cdot 5 ^ { 2 } ) ^ { 6 } \cdot ( 3 ^ { 2 } \cdot 2 ^ { 5 } ) ^ { 4 } } { ( 2 ^ { 3 } \cdot 3 \cdot 5 ^ { 2 } ) ^ { 5 } \cdot ( 2 ^ { 7 } 35 ) ^ { 3 } }$$

Evaluate

$$\frac{108}{1715}\approx 0.062973761$$

Solution Steps

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(8\times 5^{2}\right)^{6}\times \left(3^{2}\times 2^{5}\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(8\times 25\right)^{6}\times \left(3^{2}\times 2^{5}\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Multiply 8 and 25 to get 200.

$$\frac{200^{6}\times \left(3^{2}\times 2^{5}\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 200 to the power of 6 and get 64000000000000.

$$\frac{64000000000000\times \left(3^{2}\times 2^{5}\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{64000000000000\times \left(9\times 2^{5}\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{64000000000000\times \left(9\times 32\right)^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Multiply 9 and 32 to get 288.

$$\frac{64000000000000\times 288^{4}}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 288 to the power of 4 and get 6879707136.

$$\frac{64000000000000\times 6879707136}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Multiply 64000000000000 and 6879707136 to get 440301256704000000000000.

$$\frac{440301256704000000000000}{\left(2^{3}\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{440301256704000000000000}{\left(8\times 3\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Multiply 8 and 3 to get 24.

$$\frac{440301256704000000000000}{\left(24\times 5^{2}\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{440301256704000000000000}{\left(24\times 25\right)^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Multiply 24 and 25 to get 600.

$$\frac{440301256704000000000000}{600^{5}\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 600 to the power of 5 and get 77760000000000.

$$\frac{440301256704000000000000}{77760000000000\times \left(2^{7}\times 35\right)^{3}}$$

Calculate 2 to the power of 7 and get 128.

$$\frac{440301256704000000000000}{77760000000000\times \left(128\times 35\right)^{3}}$$

Multiply 128 and 35 to get 4480.

$$\frac{440301256704000000000000}{77760000000000\times 4480^{3}}$$

Calculate 4480 to the power of 3 and get 89915392000.

$$\frac{440301256704000000000000}{77760000000000\times 89915392000}$$

Multiply 77760000000000 and 89915392000 to get 6991820881920000000000000.

$$\frac{440301256704000000000000}{6991820881920000000000000}$$

Reduce the fraction \frac{440301256704000000000000}{6991820881920000000000000} to lowest terms by extracting and canceling out 4076863488000000000000.

$$\frac{108}{1715}$$

Factor

$$\frac{2 ^ {2} \cdot 3 ^ {3}}{5 \cdot 7 ^ {3}} = 0.0629737609329446$$