Solve frac{8!m^{5}m^2n^-2}{9m^-9n^3}

$$\frac { 8 ! m ^ { 5 } m ^ { 2 } n ^ { - 2 } } { 9 m ^ { - 9 } n ^ { 3 } }$$

Evaluate

$$\frac{4480m^{16}}{n^{5}}$$

Solution Steps

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

$$\frac{8!m^{7}n^{-2}}{9m^{-9}n^{3}}$$

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

$$\frac{8!n^{-2}m^{16}}{9n^{3}}$$

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

$$\frac{8!m^{16}}{9n^{5}}$$

The factorial of 8 is 40320.

$$\frac{40320m^{16}}{9n^{5}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{4480m^{16}}{n^{5}}$$

Differentiate w.r.t. n

$$-\frac{22400m^{16}}{n^{6}}$$