Solve frac{(2m^{4})^{-1}}{(m^-2)^0*m^-1n^3}

$$\frac { ( 2 m ^ { 4 } ) ^ { - 1 } } { ( m ^ { - 2 } ) ^ { 0 } \cdot m ^ { - 1 } n ^ { 3 } }$$

Evaluate

$$\frac{1}{2\left(mn\right)^{3}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -2 and 0 to get 0.

$$\frac{\left(2m^{4}\right)^{-1}}{m^{0}m^{-1}n^{3}}$$

To multiply powers of the same base, add their exponents. Add 0 and -1 to get -1.

$$\frac{\left(2m^{4}\right)^{-1}}{m^{-1}n^{3}}$$

Expand \left(2m^{4}\right)^{-1}.

$$\frac{2^{-1}\left(m^{4}\right)^{-1}}{m^{-1}n^{3}}$$

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

$$\frac{2^{-1}m^{-4}}{m^{-1}n^{3}}$$

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

$$\frac{\frac{1}{2}m^{-4}}{m^{-1}n^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\frac{1}{2}}{m^{3}n^{3}}$$

Express \frac{\frac{1}{2}}{m^{3}n^{3}} as a single fraction.

$$\frac{1}{2m^{3}n^{3}}$$

Differentiate w.r.t. n

$$-\frac{3}{2m^{3}n^{4}}$$