Solve frac{9m^{-2}(n^3)^0}{(3mn^-1)^4}

$$\frac { 9 m ^ { - 2 } ( n ^ { 3 } ) ^ { 0 } } { ( 3 m n ^ { - 1 } ) ^ { 4 } }$$

Evaluate

$$\frac{n^{4}}{9m^{6}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 0 to get 0.

$$\frac{9m^{-2}n^{0}}{\left(3mn^{-1}\right)^{4}}$$

Calculate n to the power of 0 and get 1.

$$\frac{9m^{-2}\times 1}{\left(3mn^{-1}\right)^{4}}$$

Multiply 9 and 1 to get 9.

$$\frac{9m^{-2}}{\left(3mn^{-1}\right)^{4}}$$

Expand \left(3mn^{-1}\right)^{4}.

$$\frac{9m^{-2}}{3^{4}m^{4}\left(n^{-1}\right)^{4}}$$

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

$$\frac{9m^{-2}}{3^{4}m^{4}n^{-4}}$$

Calculate 3 to the power of 4 and get 81.

$$\frac{9m^{-2}}{81m^{4}n^{-4}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{m^{-2}}{9n^{-4}m^{4}}$$

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

$$\frac{1}{9n^{-4}m^{6}}$$

Differentiate w.r.t. n

$$\frac{4n^{3}}{9m^{6}}$$