Solve (x + y)/(x - y) - (x - y)/(x + y) + (4xy)/(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

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

Evaluate

$\frac{8yx^{3}}{x^{4}-y^{4}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-y$ and $x+y$ is $\left(x+y\right)\left(x-y\right)$. Multiply $\frac{x+y}{x-y}$ times $\frac{x+y}{x+y}$. Multiply $\frac{x-y}{x+y}$ times $\frac{x-y}{x-y}$.

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

Since $\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$ and $\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $\left(x+y\right)\left(x+y\right)-\left(x-y\right)\left(x-y\right)$.

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

Combine like terms in $x^{2}+xy+xy+y^{2}-x^{2}+xy+xy-y^{2}$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+y\right)\left(x-y\right)$ and $x^{2}+y^{2}$ is $\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)$. Multiply $\frac{4xy}{\left(x+y\right)\left(x-y\right)}$ times $\frac{x^{2}+y^{2}}{x^{2}+y^{2}}$. Multiply $\frac{4xy}{x^{2}+y^{2}}$ times $\frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$.

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

Since $\frac{4xy\left(x^{2}+y^{2}\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ and $\frac{4xy\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $4xy\left(x^{2}+y^{2}\right)+4xy\left(x+y\right)\left(x-y\right)$.

$$\frac{4x^{3}y+4xy^{3}+4x^{3}y-4x^{2}y^{2}+4x^{2}y^{2}-4xy^{3}}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Combine like terms in $4x^{3}y+4xy^{3}+4x^{3}y-4x^{2}y^{2}+4x^{2}y^{2}-4xy^{3}$.

$$\frac{8x^{3}y}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

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

$$\frac{8x^{3}y}{x^{4}-y^{4}}$$

Expand

$\frac{8yx^{3}}{x^{4}-y^{4}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-y$ and $x+y$ is $\left(x+y\right)\left(x-y\right)$. Multiply $\frac{x+y}{x-y}$ times $\frac{x+y}{x+y}$. Multiply $\frac{x-y}{x+y}$ times $\frac{x-y}{x-y}$.

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

Since $\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$ and $\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $\left(x+y\right)\left(x+y\right)-\left(x-y\right)\left(x-y\right)$.

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

Combine like terms in $x^{2}+xy+xy+y^{2}-x^{2}+xy+xy-y^{2}$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+y\right)\left(x-y\right)$ and $x^{2}+y^{2}$ is $\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)$. Multiply $\frac{4xy}{\left(x+y\right)\left(x-y\right)}$ times $\frac{x^{2}+y^{2}}{x^{2}+y^{2}}$. Multiply $\frac{4xy}{x^{2}+y^{2}}$ times $\frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$.

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

Since $\frac{4xy\left(x^{2}+y^{2}\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ and $\frac{4xy\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $4xy\left(x^{2}+y^{2}\right)+4xy\left(x+y\right)\left(x-y\right)$.

$$\frac{4x^{3}y+4xy^{3}+4x^{3}y-4x^{2}y^{2}+4x^{2}y^{2}-4xy^{3}}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

Combine like terms in $4x^{3}y+4xy^{3}+4x^{3}y-4x^{2}y^{2}+4x^{2}y^{2}-4xy^{3}$.

$$\frac{8x^{3}y}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$$

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

$$\frac{8x^{3}y}{x^{4}-y^{4}}$$