Simplify (x+iy)^2

${(x + i y)}^{2}$

Rewrite ${(x + i y)}^{2}$ as $(x + i y) (x + i y)$ .

$(x + i y) (x + i y)$

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

Apply the distributive property.

$x (x + i y) + i y (x + i y)$

Apply the distributive property.

$x \cdot x + x (i y) + i y (x + i y)$

Apply the distributive property.

$x \cdot x + x (i y) + i y x + i y (i y)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (i y) + i y x + i y (i y)$

Rewrite using the commutative property of multiplication.

$x^{2} + i x y + i y x + i y (i y)$

Rewrite using the commutative property of multiplication.

$x^{2} + i x y + i y x + i \cdot i y \cdot y$

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

Move $y$ .

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

Multiply $y$ by $y$ .

$x^{2} + i x y + i y x + i \cdot i y^{2}$

Multiply $i i$ .

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

$x^{2} + i x y + i y x + i^{1} i y^{2}$

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

$x^{2} + i x y + i y x + i^{1} i^{1} y^{2}$

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

$x^{2} + i x y + i y x + i^{1 + 1} y^{2}$

Add $1$ and $1$ .

$x^{2} + i x y + i y x + i^{2} y^{2}$

Rewrite $i^{2}$ as $- 1$ .

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

Reorder $y$ and $x$ .

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

Add $x y i$ and $x y i$ .

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