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

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+\sqrt{y}\right)^{2}.

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

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

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x}-\sqrt{y}\right)^{2}.

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

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

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

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

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

Use the distributive property to multiply x+2\sqrt{x}\sqrt{y}+y by x-2\sqrt{x}\sqrt{y}+y and combine like terms.

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

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

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

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

$$x^{2}+2xy-4xy+y^{2}$$

Combine 2xy and -4xy to get -2xy.

$$x^{2}-2xy+y^{2}$$

Expand

$$x^{2}-2xy+y^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+\sqrt{y}\right)^{2}.

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

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

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x}-\sqrt{y}\right)^{2}.

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

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

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

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

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

Use the distributive property to multiply x+2\sqrt{x}\sqrt{y}+y by x-2\sqrt{x}\sqrt{y}+y and combine like terms.

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

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

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

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

$$x^{2}+2xy-4xy+y^{2}$$

Combine 2xy and -4xy to get -2xy.

$$x^{2}-2xy+y^{2}$$