Solve x^2*(x/y-y/x)*(y/x+x/y)*y^2

$$x ^ { 2 } \times ( \frac { x } { y } - \frac { y } { x } ) \times ( \frac { y } { x } + \frac { x } { y } ) \times y ^ { 2 }$$

Evaluate

$$x^{4}-y^{4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

$$x^{2}\left(\frac{xx}{xy}-\frac{yy}{xy}\right)\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

$$x^{2}\times \frac{xx-yy}{xy}\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

Do the multiplications in xx-yy.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{y}{x} times \frac{y}{y}. Multiply \frac{x}{y} times \frac{x}{x}.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\left(\frac{yy}{xy}+\frac{xx}{xy}\right)y^{2}$$

Since \frac{yy}{xy} and \frac{xx}{xy} have the same denominator, add them by adding their numerators.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\times \frac{yy+xx}{xy}y^{2}$$

Do the multiplications in yy+xx.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

Express x^{2}\times \frac{x^{2}-y^{2}}{xy} as a single fraction.

$$\frac{x^{2}\left(x^{2}-y^{2}\right)}{xy}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(x^{2}-y^{2}\right)}{y}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

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

$$\frac{x\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}\right)}{yxy}y^{2}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)}{yy}y^{2}$$

Express \frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)}{yy}y^{2} as a single fraction.

$$\frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)y^{2}}{yy}$$

Cancel out yy in both numerator and denominator.

$$\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-\left(y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-y^{4}$$

Expand

$$x^{4}-y^{4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{x}{y} times \frac{x}{x}. Multiply \frac{y}{x} times \frac{y}{y}.

$$x^{2}\left(\frac{xx}{xy}-\frac{yy}{xy}\right)\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

$$x^{2}\times \frac{xx-yy}{xy}\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

Do the multiplications in xx-yy.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\left(\frac{y}{x}+\frac{x}{y}\right)y^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and y is xy. Multiply \frac{y}{x} times \frac{y}{y}. Multiply \frac{x}{y} times \frac{x}{x}.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\left(\frac{yy}{xy}+\frac{xx}{xy}\right)y^{2}$$

Since \frac{yy}{xy} and \frac{xx}{xy} have the same denominator, add them by adding their numerators.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\times \frac{yy+xx}{xy}y^{2}$$

Do the multiplications in yy+xx.

$$x^{2}\times \frac{x^{2}-y^{2}}{xy}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

Express x^{2}\times \frac{x^{2}-y^{2}}{xy} as a single fraction.

$$\frac{x^{2}\left(x^{2}-y^{2}\right)}{xy}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

Cancel out x in both numerator and denominator.

$$\frac{x\left(x^{2}-y^{2}\right)}{y}\times \frac{y^{2}+x^{2}}{xy}y^{2}$$

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

$$\frac{x\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}\right)}{yxy}y^{2}$$

Cancel out x in both numerator and denominator.

$$\frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)}{yy}y^{2}$$

Express \frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)}{yy}y^{2} as a single fraction.

$$\frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)y^{2}}{yy}$$

Cancel out yy in both numerator and denominator.

$$\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-\left(y^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-y^{4}$$