Solve (x-y)(x+y)((x)^(2)+(y)^(2))(((x)^(2)+(y)^(2))^(2)-2(x)^(2)(y)^(2)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$x^{8}-y^{8}$

Solution Steps

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

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

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

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

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

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

Combine $2x^{2}y^{2}$ and $-2x^{2}y^{2}$ to get $0$.

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

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$x^{8}-\left(y^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$x^{8}-y^{8}$$

Expand

$x^{8}-y^{8}$

Solution Steps

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

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

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

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

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

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

Combine $2x^{2}y^{2}$ and $-2x^{2}y^{2}$ to get $0$.

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

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

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

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

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

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

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

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$x^{8}-\left(y^{4}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply $4$ and $2$ to get $8$.

$$x^{8}-y^{8}$$