Solve (sqrt{x}-sqrt{y})(sqrt{x}+sqrt{y})(x+y)(x^2+y^2)

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

Evaluate

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

Solution Steps

Use the distributive property to multiply \sqrt{x}-\sqrt{y} by \sqrt{x}+\sqrt{y} and combine like terms.

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

Calculate \sqrt{x} to the power of 2 and get x.

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

Calculate \sqrt{y} to the power of 2 and get y.

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

Use the distributive property to multiply x-y by x+y and combine like terms.

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

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.

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

Differentiate w.r.t. x

$$4x^{3}$$