Solve m^19n^19(1/m)^30(1/n)^30

$$m ^ { 19 } n ^ { 19 } ( \frac { 1 } { m } ) ^ { 30 } ( \frac { 1 } { n } ) ^ { 30 }$$

Evaluate

$$\frac{1}{\left(mn\right)^{11}}$$

Solution Steps

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

$$m^{19}n^{19}\times \frac{1^{30}}{m^{30}}\times \left(\frac{1}{n}\right)^{30}$$

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

$$m^{19}n^{19}\times \frac{1^{30}}{m^{30}}\times \frac{1^{30}}{n^{30}}$$

Express m^{19}\times \frac{1^{30}}{m^{30}} as a single fraction.

$$\frac{m^{19}\times 1^{30}}{m^{30}}n^{19}\times \frac{1^{30}}{n^{30}}$$

Cancel out m^{19} in both numerator and denominator.

$$\frac{1^{30}}{m^{11}}n^{19}\times \frac{1^{30}}{n^{30}}$$

Multiply \frac{1^{30}}{m^{11}} times \frac{1^{30}}{n^{30}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1^{30}\times 1^{30}}{m^{11}n^{30}}n^{19}$$

To multiply powers of the same base, add their exponents. Add 30 and 30 to get 60.

$$\frac{1^{60}}{m^{11}n^{30}}n^{19}$$

Calculate 1 to the power of 60 and get 1.

$$\frac{1}{m^{11}n^{30}}n^{19}$$

Express \frac{1}{m^{11}n^{30}}n^{19} as a single fraction.

$$\frac{n^{19}}{m^{11}n^{30}}$$

Cancel out n^{19} in both numerator and denominator.

$$\frac{1}{m^{11}n^{11}}$$

Differentiate w.r.t. m

$$-\frac{11}{n^{11}m^{12}}$$