Solve frac{(2+2)^{2}}{2^{2}}*(3+3+3+3)^3/(3+3+3)^3*(6+6+6+6+6+6)^6/(6+6+6+6)^6

$$\frac { ( 2 + 2 ) ^ { 2 } } { 2 ^ { 2 } } \cdot \frac { ( 3 + 3 + 3 + 3 ) ^ { 3 } } { ( 3 + 3 + 3 ) ^ { 3 } } \cdot \frac { ( 6 + 6 + 6 + 6 + 6 + 6 ) ^ { 6 } } { ( 6 + 6 + 6 + 6 ) ^ { 6 } }$$

Evaluate

$$108$$

Solution Steps

Add 2 and 2 to get 4.

$$\frac{4^{2}}{2^{2}}\times \frac{\left(3+3+3+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16}{2^{2}}\times \frac{\left(3+3+3+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{16}{4}\times \frac{\left(3+3+3+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Divide 16 by 4 to get 4.

$$4\times \frac{\left(3+3+3+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 3 and 3 to get 6.

$$4\times \frac{\left(6+3+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 6 and 3 to get 9.

$$4\times \frac{\left(9+3\right)^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 9 and 3 to get 12.

$$4\times \frac{12^{3}}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Calculate 12 to the power of 3 and get 1728.

$$4\times \frac{1728}{\left(3+3+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 3 and 3 to get 6.

$$4\times \frac{1728}{\left(6+3\right)^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 6 and 3 to get 9.

$$4\times \frac{1728}{9^{3}}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Calculate 9 to the power of 3 and get 729.

$$4\times \frac{1728}{729}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Reduce the fraction \frac{1728}{729} to lowest terms by extracting and canceling out 27.

$$4\times \frac{64}{27}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Multiply 4 and \frac{64}{27} to get \frac{256}{27}.

$$\frac{256}{27}\times \frac{\left(6+6+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 6 and 6 to get 12.

$$\frac{256}{27}\times \frac{\left(12+6+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 12 and 6 to get 18.

$$\frac{256}{27}\times \frac{\left(18+6+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 18 and 6 to get 24.

$$\frac{256}{27}\times \frac{\left(24+6+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 24 and 6 to get 30.

$$\frac{256}{27}\times \frac{\left(30+6\right)^{6}}{\left(6+6+6+6\right)^{6}}$$

Add 30 and 6 to get 36.

$$\frac{256}{27}\times \frac{36^{6}}{\left(6+6+6+6\right)^{6}}$$

Calculate 36 to the power of 6 and get 2176782336.

$$\frac{256}{27}\times \frac{2176782336}{\left(6+6+6+6\right)^{6}}$$

Add 6 and 6 to get 12.

$$\frac{256}{27}\times \frac{2176782336}{\left(12+6+6\right)^{6}}$$

Add 12 and 6 to get 18.

$$\frac{256}{27}\times \frac{2176782336}{\left(18+6\right)^{6}}$$

Add 18 and 6 to get 24.

$$\frac{256}{27}\times \frac{2176782336}{24^{6}}$$

Calculate 24 to the power of 6 and get 191102976.

$$\frac{256}{27}\times \frac{2176782336}{191102976}$$

Reduce the fraction \frac{2176782336}{191102976} to lowest terms by extracting and canceling out 2985984.

$$\frac{256}{27}\times \frac{729}{64}$$

Multiply \frac{256}{27} and \frac{729}{64} to get 108.

$$108$$

Factor

$$2^{2}\times 3^{3}$$