Solve -[(m)^3:(1/n)^-2]^-1*m^-5

$$- [ ( m ) ^ { 3 } : ( \frac { 1 } { n } ) ^ { - 2 } ] ^ { - 1 } \cdot m ^ { - 5 }$$

Evaluate

$$-\frac{n^{2}}{m^{8}}$$

Solution Steps

To raise \frac{1}{n} to a power, raise both numerator and denominator to the power and then divide.

$$\left(-\left(\frac{m^{3}}{\frac{1^{-2}}{n^{-2}}}\right)^{-1}\right)m^{-5}$$

Divide m^{3} by \frac{1^{-2}}{n^{-2}} by multiplying m^{3} by the reciprocal of \frac{1^{-2}}{n^{-2}}.

$$\left(-\left(\frac{m^{3}n^{-2}}{1^{-2}}\right)^{-1}\right)m^{-5}$$

Calculate 1 to the power of -2 and get 1.

$$\left(-\left(\frac{m^{3}n^{-2}}{1}\right)^{-1}\right)m^{-5}$$

Anything divided by one gives itself.

$$\left(-\left(m^{3}n^{-2}\right)^{-1}\right)m^{-5}$$

Expand \left(m^{3}n^{-2}\right)^{-1}.

$$\left(-\left(m^{3}\right)^{-1}\left(n^{-2}\right)^{-1}\right)m^{-5}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\left(-m^{-3}\left(n^{-2}\right)^{-1}\right)m^{-5}$$

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

$$\left(-m^{-3}n^{2}\right)m^{-5}$$

To multiply powers of the same base, add their exponents. Add -3 and -5 to get -8.

$$-m^{-8}n^{2}$$

Differentiate w.r.t. n

$$-\frac{2n}{m^{8}}$$