Solve {[(1/4)^{2}*(2/5)^{3}]^2*(4/5)^5}:(8/25)^5*frac{1}{2^3}]

$$\{ [ ( \frac { 1 } { 4 } ) ^ { 2 } \cdot ( \frac { 2 } { 5 } ) ^ { 3 } ] ^ { 2 } \cdot ( \frac { 4 } { 5 } ) ^ { 5 } \} : ( \frac { 8 } { 25 } ) ^ { 5 } \cdot \frac { 1 } { 2 ^ { 3 } } ]$$

Evaluate

$$\frac{1}{5120}=0.000195312$$

Solution Steps

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

$$\frac{\left(\frac{1}{16}\times \left(\frac{2}{5}\right)^{3}\right)^{2}\times \left(\frac{4}{5}\right)^{5}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Calculate \frac{2}{5} to the power of 3 and get \frac{8}{125}.

$$\frac{\left(\frac{1}{16}\times \frac{8}{125}\right)^{2}\times \left(\frac{4}{5}\right)^{5}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Multiply \frac{1}{16} and \frac{8}{125} to get \frac{1}{250}.

$$\frac{\left(\frac{1}{250}\right)^{2}\times \left(\frac{4}{5}\right)^{5}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Calculate \frac{1}{250} to the power of 2 and get \frac{1}{62500}.

$$\frac{\frac{1}{62500}\times \left(\frac{4}{5}\right)^{5}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Calculate \frac{4}{5} to the power of 5 and get \frac{1024}{3125}.

$$\frac{\frac{1}{62500}\times \frac{1024}{3125}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Multiply \frac{1}{62500} and \frac{1024}{3125} to get \frac{256}{48828125}.

$$\frac{\frac{256}{48828125}}{\left(\frac{8}{25}\right)^{5}}\times \frac{1}{2^{3}}$$

Calculate \frac{8}{25} to the power of 5 and get \frac{32768}{9765625}.

$$\frac{\frac{256}{48828125}}{\frac{32768}{9765625}}\times \frac{1}{2^{3}}$$

Divide \frac{256}{48828125} by \frac{32768}{9765625} by multiplying \frac{256}{48828125} by the reciprocal of \frac{32768}{9765625}.

$$\frac{256}{48828125}\times \frac{9765625}{32768}\times \frac{1}{2^{3}}$$

Multiply \frac{256}{48828125} and \frac{9765625}{32768} to get \frac{1}{640}.

$$\frac{1}{640}\times \frac{1}{2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{1}{640}\times \frac{1}{8}$$

Multiply \frac{1}{640} and \frac{1}{8} to get \frac{1}{5120}.

$$\frac{1}{5120}$$

Factor

$$\frac{1}{2 ^ {10} \cdot 5} = 0.0001953125$$