Solve (frac{5^{4}}{2^{-3}})^{6}(frac{5^{-4}}{5^{-1}})^{-5}(frac{2^{-2}}{5^{-2}})^3(2^3/2^-1)^-1

$$( \frac { 5 ^ { 4 } } { 2 ^ { - 3 } } ) ^ { 6 } ( \frac { 5 ^ { - 4 } } { 5 ^ { - 1 } } ) ^ { - 5 } ( \frac { 2 ^ { - 2 } } { 5 ^ { - 2 } } ) ^ { 3 } ( \frac { 2 ^ { 3 } } { 2 ^ { - 1 } } ) ^ { - 1 }$$

Evaluate

$$7275957614183425903320312500000000$$

Solution Steps

Rewrite 5^{-1} as 5^{-4}\times 5^{3}. Cancel out 5^{-4} in both numerator and denominator.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -1 from 3 to get 4.

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

To raise a power to another power, multiply the exponents. Multiply 4 and -1 to get -4.

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

Calculate 5 to the power of 4 and get 625.

$$\left(\frac{625}{2^{-3}}\right)^{6}\times \left(\frac{1}{5^{3}}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\left(\frac{625}{\frac{1}{8}}\right)^{6}\times \left(\frac{1}{5^{3}}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Divide 625 by \frac{1}{8} by multiplying 625 by the reciprocal of \frac{1}{8}.

$$\left(625\times 8\right)^{6}\times \left(\frac{1}{5^{3}}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Multiply 625 and 8 to get 5000.

$$5000^{6}\times \left(\frac{1}{5^{3}}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Calculate 5000 to the power of 6 and get 15625000000000000000000.

$$15625000000000000000000\times \left(\frac{1}{5^{3}}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Calculate 5 to the power of 3 and get 125.

$$15625000000000000000000\times \left(\frac{1}{125}\right)^{-5}\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Calculate \frac{1}{125} to the power of -5 and get 30517578125.

$$15625000000000000000000\times 30517578125\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Multiply 15625000000000000000000 and 30517578125 to get 476837158203125000000000000000000.

$$476837158203125000000000000000000\times \left(\frac{2^{-2}}{5^{-2}}\right)^{3}\times 2^{-4}$$

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

$$476837158203125000000000000000000\times \left(\frac{\frac{1}{4}}{5^{-2}}\right)^{3}\times 2^{-4}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$476837158203125000000000000000000\times \left(\frac{\frac{1}{4}}{\frac{1}{25}}\right)^{3}\times 2^{-4}$$

Divide \frac{1}{4} by \frac{1}{25} by multiplying \frac{1}{4} by the reciprocal of \frac{1}{25}.

$$476837158203125000000000000000000\times \left(\frac{1}{4}\times 25\right)^{3}\times 2^{-4}$$

Multiply \frac{1}{4} and 25 to get \frac{25}{4}.

$$476837158203125000000000000000000\times \left(\frac{25}{4}\right)^{3}\times 2^{-4}$$

Calculate \frac{25}{4} to the power of 3 and get \frac{15625}{64}.

$$476837158203125000000000000000000\times \frac{15625}{64}\times 2^{-4}$$

Multiply 476837158203125000000000000000000 and \frac{15625}{64} to get 116415321826934814453125000000000000.

$$116415321826934814453125000000000000\times 2^{-4}$$

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$116415321826934814453125000000000000\times \frac{1}{16}$$

Multiply 116415321826934814453125000000000000 and \frac{1}{16} to get 7275957614183425903320312500000000.

$$7275957614183425903320312500000000$$

Factor

$$2^{8}\times 5^{45}$$