Solve (2m^{frac{1/3}n^{1/6})^{0}}{(2m^-2n^6)^-1(2mn)^5}

$$\frac { ( 2 m ^ { \frac { 1 } { 3 } } n ^ { \frac { 1 } { 6 } } ) ^ { 0 } } { ( 2 m ^ { - 2 } n ^ { 6 } ) ^ { - 1 } ( 2 m n ) ^ { 5 } }$$

Evaluate

$$\frac{n}{16m^{7}}$$

Solution Steps

Calculate 2m^{\frac{1}{3}}n^{\frac{1}{6}} to the power of 0 and get 1.

$$\frac{1}{\left(2m^{-2}n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

Expand \left(2m^{-2}n^{6}\right)^{-1}.

$$\frac{1}{2^{-1}\left(m^{-2}\right)^{-1}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -1 to get 2.

$$\frac{1}{2^{-1}m^{2}\left(n^{6}\right)^{-1}\times \left(2mn\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -1 to get -6.

$$\frac{1}{2^{-1}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

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

$$\frac{1}{\frac{1}{2}m^{2}n^{-6}\times \left(2mn\right)^{5}}$$

Expand \left(2mn\right)^{5}.

$$\frac{1}{\frac{1}{2}m^{2}n^{-6}\times 2^{5}m^{5}n^{5}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{1}{\frac{1}{2}m^{2}n^{-6}\times 32m^{5}n^{5}}$$

Multiply \frac{1}{2} and 32 to get 16.

$$\frac{1}{16m^{2}n^{-6}m^{5}n^{5}}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{1}{16m^{7}n^{-6}n^{5}}$$

To multiply powers of the same base, add their exponents. Add -6 and 5 to get -1.

$$\frac{1}{16m^{7}n^{-1}}$$

Differentiate w.r.t. m

$$-\frac{7n}{16m^{8}}$$