Solve (frac{5m^{-6}n^{-2}p^-1}{10m^-8np^-7})^-1

$$( \frac { 5 m ^ { - 6 } n ^ { - 2 } p ^ { - 1 } } { 10 m ^ { - 8 } n p ^ { - 7 } } ) ^ { - 1 }$$

Evaluate

$$\frac{2n^{3}}{m^{2}p^{6}}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

$$\left(\frac{m^{-6}n^{-2}\times \frac{1}{p}}{2m^{-8}p^{-7}n}\right)^{-1}$$

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

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

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

$$\left(\frac{m^{2}p^{6}}{2n^{3}}\right)^{-1}$$

To raise \frac{m^{2}p^{6}}{2n^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(m^{2}p^{6}\right)^{-1}}{\left(2n^{3}\right)^{-1}}$$

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

$$\frac{\left(m^{2}\right)^{-1}\left(p^{6}\right)^{-1}}{\left(2n^{3}\right)^{-1}}$$

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

$$\frac{m^{-2}\left(p^{6}\right)^{-1}}{\left(2n^{3}\right)^{-1}}$$

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

$$\frac{m^{-2}p^{-6}}{\left(2n^{3}\right)^{-1}}$$

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

$$\frac{m^{-2}p^{-6}}{2^{-1}\left(n^{3}\right)^{-1}}$$

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

$$\frac{m^{-2}p^{-6}}{2^{-1}n^{-3}}$$

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

$$\frac{m^{-2}p^{-6}}{\frac{1}{2}n^{-3}}$$

Differentiate w.r.t. p

$$-\frac{12n^{3}}{m^{2}p^{7}}$$