Simplify (5+9i)^2

${(5 + 9 i)}^{2}$

Step 1

Rewrite ${(5 + 9 i)}^{2}$ as $(5 + 9 i) (5 + 9 i)$ .

$(5 + 9 i) (5 + 9 i)$

Step 2

Expand $(5 + 9 i) (5 + 9 i)$ using the FOIL Method.

Apply the distributive property.

$5 (5 + 9 i) + 9 i (5 + 9 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (9 i) + 9 i (5 + 9 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (9 i) + 9 i \cdot 5 + 9 i (9 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (9 i) + 9 i \cdot 5 + 9 i (9 i)$

Multiply $9$ by $5$ .

$25 + 45 i + 9 i \cdot 5 + 9 i (9 i)$

Multiply $5$ by $9$ .

$25 + 45 i + 45 i + 9 i (9 i)$

Multiply $9 i (9 i)$ .

Multiply $9$ by $9$ .

$25 + 45 i + 45 i + 81 i i$

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

$25 + 45 i + 45 i + 81 (i^{1} i)$

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

$25 + 45 i + 45 i + 81 (i^{1} i^{1})$

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

$25 + 45 i + 45 i + 81 i^{1 + 1}$

Add $1$ and $1$ .

$25 + 45 i + 45 i + 81 i^{2}$

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

$25 + 45 i + 45 i + 81 \cdot - 1$

Multiply $81$ by $- 1$ .

$25 + 45 i + 45 i - 81$

Subtract $81$ from $25$ .

$- 56 + 45 i + 45 i$

Add $45 i$ and $45 i$ .

$- 56 + 90 i$