Simplify x/(x^2-y^2)+(x+y)/(x^2+y^2)

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

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

To write $\frac{x}{(x + y) (x - y)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

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

To write $\frac{x + y}{x^{2} + y^{2}}$ as a fraction with a common denominator, multiply by $\frac{(x + y) (x - y)}{(x + y) (x - y)}$ .

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

Write each expression with a common denominator of $(x + y) (x - y) (x^{2} + y^{2})$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{(x + y) (x - y)}$ by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

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

Multiply $\frac{x + y}{x^{2} + y^{2}}$ by $\frac{(x + y) (x - y)}{(x + y) (x - y)}$ .

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

Reorder the factors of $(x^{2} + y^{2}) ((x + y) (x - y))$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{2} + x y^{2} + (x + y) (x + y) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 2} + x y^{2} + (x + y) (x + y) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Add $1$ and $2$ .

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

Expand $(x + y) (x + y)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$\frac{x^{3} + x y^{2} + (x \cdot x + x y + y (x + y)) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Apply the distributive property.

$\frac{x^{3} + x y^{2} + (x \cdot x + x y + y x + y \cdot y) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{3} + x y^{2} + (x^{2} + x y + y x + y \cdot y) (x - y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $y$ by $y$ .

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

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

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

Add $x y$ and $x y$ .

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

Expand $(x^{2} + 2 x y + y^{2}) (x - y)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$\frac{x^{3} + x y^{2} + x^{2} x^{1} + x^{2} (- y) + 2 x y x + 2 x y (- y) + y^{2} x + y^{2} (- y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + x y^{2} + x^{2 + 1} + x^{2} (- y) + 2 x y x + 2 x y (- y) + y^{2} x + y^{2} (- y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 (x \cdot x) y + 2 x y (- y) + y^{2} x + y^{2} (- y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $x$ by $x$ .

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 x^{2} y + 2 x (y \cdot y) \cdot - 1 + y^{2} x + y^{2} (- y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $y$ by $y$ .

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 x^{2} y + 2 x y^{2} \cdot - 1 + y^{2} x + y^{2} (- y)}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $- 1$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 x^{2} y - 2 x y^{2} + y^{2} x - (y \cdot y^{2})}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $y$ by $y^{2}$ .

Raise $y$ to the power of $1$ .

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 x^{2} y - 2 x y^{2} + y^{2} x - (y^{1} y^{2})}{(x + y) (x - y) (x^{2} + y^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + x y^{2} + x^{3} - x^{2} y + 2 x^{2} y - 2 x y^{2} + y^{2} x - y^{1 + 2}}{(x + y) (x - y) (x^{2} + y^{2})}$

Add $1$ and $2$ .

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

Add $- x^{2} y$ and $2 x^{2} y$ .

$\frac{x^{3} + x y^{2} + x^{3} + 1 x^{2} y - 2 x y^{2} + y^{2} x - y^{3}}{(x + y) (x - y) (x^{2} + y^{2})}$

Multiply $x^{2}$ by $1$ .

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

Add $- 2 x y^{2}$ and $y^{2} x$ .

Reorder $y^{2}$ and $x$ .

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

Add $- 2 x y^{2}$ and $x y^{2}$ .

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

Add $x^{3}$ and $x^{3}$ .

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

Subtract $x y^{2}$ from $x y^{2}$ .

$\frac{2 x^{3} + x^{2} y + 0 - y^{3}}{(x + y) (x - y) (x^{2} + y^{2})}$

Add $2 x^{3}$ and $0$ .

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

Simplify x/(x^2-y^2)+(x+y)/(x^2+y^2)

Related Questions

Support LCMGCF.com by disabling your adblocker.