Solve x^4-y^4=(x-y)(x+y)(x^2+y^2)

$$x ^ { 4 } - y ^ { 4 } = ( x - y ) ( x + y ) ( x ^ { 2 } + y ^ { 2 } )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

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

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

Consider \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}.

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

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

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

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

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

Subtract x^{4} from both sides.

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

Combine x^{4} and -x^{4} to get 0.

$$-y^{4}=-y^{4}$$

Cancel out -1 on both sides.

$$y^{4}=y^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

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

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

Consider \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}.

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

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

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

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

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

Add y^{4} to both sides.

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

Combine -y^{4} and y^{4} to get 0.

$$x^{4}=x^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

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

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

Consider \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}.

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

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

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

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

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

Subtract x^{4} from both sides.

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

Combine x^{4} and -x^{4} to get 0.

$$-y^{4}=-y^{4}$$

Cancel out -1 on both sides.

$$y^{4}=y^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

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

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

Consider \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}.

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

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

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

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

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

Add y^{4} to both sides.

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

Combine -y^{4} and y^{4} to get 0.

$$x^{4}=x^{4}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$