Solve ((frac{x^{5}y^{2}}{x^{2}y^{5}})^{2}(frac{x^{2}y^{5}}{x^5y^2})^3)^2

$$( ( \frac { x ^ { 5 } y ^ { 2 } } { x ^ { 2 } y ^ { 5 } } ) ^ { 2 } ( \frac { x ^ { 2 } y ^ { 5 } } { x ^ { 5 } y ^ { 2 } } ) ^ { 3 } ) ^ { 2 }$$

Evaluate

$$\left(\frac{y}{x}\right)^{6}$$

Solution Steps

Cancel out x^{2}y^{2} in both numerator and denominator.

$$\left(\left(\frac{x^{3}}{y^{3}}\right)^{2}\times \left(\frac{x^{2}y^{5}}{x^{5}y^{2}}\right)^{3}\right)^{2}$$

To raise \frac{x^{3}}{y^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \left(\frac{x^{2}y^{5}}{x^{5}y^{2}}\right)^{3}\right)^{2}$$

Cancel out x^{2}y^{2} in both numerator and denominator.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \left(\frac{y^{3}}{x^{3}}\right)^{3}\right)^{2}$$

To raise \frac{y^{3}}{x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \frac{\left(y^{3}\right)^{3}}{\left(x^{3}\right)^{3}}\right)^{2}$$

Multiply \frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}} times \frac{\left(y^{3}\right)^{3}}{\left(x^{3}\right)^{3}} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{\left(x^{3}\right)^{2}\left(y^{3}\right)^{3}}{\left(y^{3}\right)^{2}\left(x^{3}\right)^{3}}\right)^{2}$$

Cancel out \left(x^{3}\right)^{2}\left(y^{3}\right)^{2} in both numerator and denominator.

$$\left(\frac{y^{3}}{x^{3}}\right)^{2}$$

To raise \frac{y^{3}}{x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(y^{3}\right)^{2}}{\left(x^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{y^{6}}{\left(x^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{y^{6}}{x^{6}}$$

Expand

$$\left(\frac{y}{x}\right)^{6}$$

Solution Steps

Cancel out x^{2}y^{2} in both numerator and denominator.

$$\left(\left(\frac{x^{3}}{y^{3}}\right)^{2}\times \left(\frac{x^{2}y^{5}}{x^{5}y^{2}}\right)^{3}\right)^{2}$$

To raise \frac{x^{3}}{y^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \left(\frac{x^{2}y^{5}}{x^{5}y^{2}}\right)^{3}\right)^{2}$$

Cancel out x^{2}y^{2} in both numerator and denominator.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \left(\frac{y^{3}}{x^{3}}\right)^{3}\right)^{2}$$

To raise \frac{y^{3}}{x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}}\times \frac{\left(y^{3}\right)^{3}}{\left(x^{3}\right)^{3}}\right)^{2}$$

Multiply \frac{\left(x^{3}\right)^{2}}{\left(y^{3}\right)^{2}} times \frac{\left(y^{3}\right)^{3}}{\left(x^{3}\right)^{3}} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{\left(x^{3}\right)^{2}\left(y^{3}\right)^{3}}{\left(y^{3}\right)^{2}\left(x^{3}\right)^{3}}\right)^{2}$$

Cancel out \left(x^{3}\right)^{2}\left(y^{3}\right)^{2} in both numerator and denominator.

$$\left(\frac{y^{3}}{x^{3}}\right)^{2}$$

To raise \frac{y^{3}}{x^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(y^{3}\right)^{2}}{\left(x^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{y^{6}}{\left(x^{3}\right)^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{y^{6}}{x^{6}}$$