Simplify ((3+6i)^2)/(2i)

$\frac{{(3 + 6 i)}^{2}}{2 i}$

Step 1

Multiply the numerator and denominator of $\frac{{(3 + 6 i)}^{2}}{2 i}$ by the conjugate of $2 i$ to make the denominator real.

$\frac{{(3 + 6 i)}^{2}}{2 i} \cdot \frac{i}{i}$

Step 2

Multiply.

Combine.

$\frac{{(3 + 6 i)}^{2} i}{2 i i}$

Simplify the numerator.

Rewrite ${(3 + 6 i)}^{2}$ as $(3 + 6 i) (3 + 6 i)$ .

$\frac{(3 + 6 i) (3 + 6 i) i}{2 i i}$

Expand $(3 + 6 i) (3 + 6 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{(3 (3 + 6 i) + 6 i (3 + 6 i)) i}{2 i i}$

Apply the distributive property.

$\frac{(3 \cdot 3 + 3 (6 i) + 6 i (3 + 6 i)) i}{2 i i}$

Apply the distributive property.

$\frac{(3 \cdot 3 + 3 (6 i) + 6 i \cdot 3 + 6 i (6 i)) i}{2 i i}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$\frac{(9 + 3 (6 i) + 6 i \cdot 3 + 6 i (6 i)) i}{2 i i}$

Multiply $6$ by $3$ .

$\frac{(9 + 18 i + 6 i \cdot 3 + 6 i (6 i)) i}{2 i i}$

Multiply $3$ by $6$ .

$\frac{(9 + 18 i + 18 i + 6 i (6 i)) i}{2 i i}$

Multiply $6 i (6 i)$ .

Multiply $6$ by $6$ .

$\frac{(9 + 18 i + 18 i + 36 i i) i}{2 i i}$

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

$\frac{(9 + 18 i + 18 i + 36 (i^{1} i)) i}{2 i i}$

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

$\frac{(9 + 18 i + 18 i + 36 (i^{1} i^{1})) i}{2 i i}$

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

$\frac{(9 + 18 i + 18 i + 36 i^{1 + 1}) i}{2 i i}$

Add $1$ and $1$ .

$\frac{(9 + 18 i + 18 i + 36 i^{2}) i}{2 i i}$

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

$\frac{(9 + 18 i + 18 i + 36 \cdot - 1) i}{2 i i}$

Multiply $36$ by $- 1$ .

$\frac{(9 + 18 i + 18 i - 36) i}{2 i i}$

Subtract $36$ from $9$ .

$\frac{(- 27 + 18 i + 18 i) i}{2 i i}$

Add $18 i$ and $18 i$ .

$\frac{(- 27 + 36 i) i}{2 i i}$

Apply the distributive property.

$\frac{- 27 i + 36 i i}{2 i i}$

Multiply $36 i i$ .

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

$\frac{- 27 i + 36 (i^{1} i)}{2 i i}$

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

$\frac{- 27 i + 36 (i^{1} i^{1})}{2 i i}$

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

$\frac{- 27 i + 36 i^{1 + 1}}{2 i i}$

Add $1$ and $1$ .

$\frac{- 27 i + 36 i^{2}}{2 i i}$

Simplify each term.

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

$\frac{- 27 i + 36 \cdot - 1}{2 i i}$

Multiply $36$ by $- 1$ .

$\frac{- 27 i - 36}{2 i i}$

Reorder $- 27 i$ and $- 36$ .

$\frac{- 36 - 27 i}{2 i i}$

Simplify the denominator.

Add parentheses.

$\frac{- 36 - 27 i}{2 (i i)}$

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

$\frac{- 36 - 27 i}{2 (i^{1} i)}$

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

$\frac{- 36 - 27 i}{2 (i^{1} i^{1})}$

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

$\frac{- 36 - 27 i}{2 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 36 - 27 i}{2 i^{2}}$

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

$\frac{- 36 - 27 i}{2 \cdot - 1}$

Step 3

Multiply $2$ by $- 1$ .

$\frac{- 36 - 27 i}{- 2}$

Step 4

Split the fraction $\frac{- 36 - 27 i}{- 2}$ into two fractions.

$\frac{- 36}{- 2} + \frac{- 27 i}{- 2}$

Step 5

Simplify each term.

Divide $- 36$ by $- 2$ .

$18 + \frac{- 27 i}{- 2}$

Dividing two negative values results in a positive value.

$18 + \frac{27 i}{2}$

Simplify ((3+6i)^2)/(2i)

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