Solve (8^{5}*5^{5}*10^{5})^{2}:(20^{2}*2^{2}*5^2)^5:(2^2*2^3)

$$( 8 ^ { 5 } \cdot 5 ^ { 5 } \cdot 10 ^ { 5 } ) ^ { 2 } : ( 20 ^ { 2 } \cdot 2 ^ { 2 } \cdot 5 ^ { 2 } ) ^ { 5 } : ( 2 ^ { 2 } \cdot 2 ^ { 3 } )$$

Evaluate

$$32$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\frac{\left(8^{5}\times 5^{5}\times 10^{5}\right)^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 8 to the power of 5 and get 32768.

$$\frac{\frac{\left(32768\times 5^{5}\times 10^{5}\right)^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 5 to the power of 5 and get 3125.

$$\frac{\frac{\left(32768\times 3125\times 10^{5}\right)^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Multiply 32768 and 3125 to get 102400000.

$$\frac{\frac{\left(102400000\times 10^{5}\right)^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{\frac{\left(102400000\times 100000\right)^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Multiply 102400000 and 100000 to get 10240000000000.

$$\frac{\frac{10240000000000^{2}}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 10240000000000 to the power of 2 and get 104857600000000000000000000.

$$\frac{\frac{104857600000000000000000000}{\left(20^{2}\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{\frac{104857600000000000000000000}{\left(400\times 2^{2}\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{104857600000000000000000000}{\left(400\times 4\times 5^{2}\right)^{5}}}{2^{5}}$$

Multiply 400 and 4 to get 1600.

$$\frac{\frac{104857600000000000000000000}{\left(1600\times 5^{2}\right)^{5}}}{2^{5}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\frac{104857600000000000000000000}{\left(1600\times 25\right)^{5}}}{2^{5}}$$

Multiply 1600 and 25 to get 40000.

$$\frac{\frac{104857600000000000000000000}{40000^{5}}}{2^{5}}$$

Calculate 40000 to the power of 5 and get 102400000000000000000000.

$$\frac{\frac{104857600000000000000000000}{102400000000000000000000}}{2^{5}}$$

Divide 104857600000000000000000000 by 102400000000000000000000 to get 1024.

$$\frac{1024}{2^{5}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{1024}{32}$$

Divide 1024 by 32 to get 32.

$$32$$

Factor

$$2^{5}$$