Solve x^19y^19{left(1/xright)}^3{left(1/yright)}^3

$${ x }^{ 19 } { y }^{ 19 } { \left( \frac{ 1 }{ x } \right) }^{ 3 } { \left( \frac{ 1 }{ y } \right) }^{ 3 }$$

Evaluate

$$\left(xy\right)^{16}$$

Short Solution Steps

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

$$x^{19}y^{19}\times \frac{1^{3}}{x^{3}}\times \left(\frac{1}{y}\right)^{3}$$

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

$$x^{19}y^{19}\times \frac{1^{3}}{x^{3}}\times \frac{1^{3}}{y^{3}}$$

Express x^{19}\times \frac{1^{3}}{x^{3}} as a single fraction.

$$\frac{x^{19}\times 1^{3}}{x^{3}}y^{19}\times \frac{1^{3}}{y^{3}}$$

Cancel out x^{3} in both numerator and denominator.

$$1^{3}x^{16}y^{19}\times \frac{1^{3}}{y^{3}}$$

Express 1^{3}x^{16}\times \frac{1^{3}}{y^{3}} as a single fraction.

$$\frac{1^{3}x^{16}\times 1^{3}}{y^{3}}y^{19}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{1^{6}x^{16}}{y^{3}}y^{19}$$

Calculate 1 to the power of 6 and get 1.

$$\frac{1x^{16}}{y^{3}}y^{19}$$

Express \frac{1x^{16}}{y^{3}}y^{19} as a single fraction.

$$\frac{1x^{16}y^{19}}{y^{3}}$$

Cancel out y^{3} in both numerator and denominator.

$$x^{16}y^{16}$$

Differentiate w.r.t. x

$$16x^{15}y^{16}$$