Write in Standard Form (5i)/(2+2i)

$\frac{5 i}{2 + 2 i}$

Step 1

Multiply the numerator and denominator of $\frac{5 i}{2 + 2 i}$ by the conjugate of $2 + 2 i$ to make the denominator real.

$\frac{5 i}{2 + 2 i} \cdot \frac{2 - 2 i}{2 - 2 i}$

Step 2

Multiply.

Combine.

$\frac{5 i (2 - 2 i)}{(2 + 2 i) (2 - 2 i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{5 i \cdot 2 + 5 i (- 2 i)}{(2 + 2 i) (2 - 2 i)}$

Multiply $2$ by $5$ .

$\frac{10 i + 5 i (- 2 i)}{(2 + 2 i) (2 - 2 i)}$

Multiply $5 i (- 2 i)$ .

Multiply $- 2$ by $5$ .

$\frac{10 i - 10 i i}{(2 + 2 i) (2 - 2 i)}$

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

$\frac{10 i - 10 (i^{1} i)}{(2 + 2 i) (2 - 2 i)}$

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

$\frac{10 i - 10 (i^{1} i^{1})}{(2 + 2 i) (2 - 2 i)}$

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

$\frac{10 i - 10 i^{1 + 1}}{(2 + 2 i) (2 - 2 i)}$

Add $1$ and $1$ .

$\frac{10 i - 10 i^{2}}{(2 + 2 i) (2 - 2 i)}$

Simplify each term.

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

$\frac{10 i - 10 \cdot - 1}{(2 + 2 i) (2 - 2 i)}$

Multiply $- 10$ by $- 1$ .

$\frac{10 i + 10}{(2 + 2 i) (2 - 2 i)}$

Reorder $10 i$ and $10$ .

$\frac{10 + 10 i}{(2 + 2 i) (2 - 2 i)}$

Simplify the denominator.

Expand $(2 + 2 i) (2 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{10 + 10 i}{2 (2 - 2 i) + 2 i (2 - 2 i)}$

Apply the distributive property.

$\frac{10 + 10 i}{2 \cdot 2 + 2 (- 2 i) + 2 i (2 - 2 i)}$

Apply the distributive property.

$\frac{10 + 10 i}{2 \cdot 2 + 2 (- 2 i) + 2 i \cdot 2 + 2 i (- 2 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{10 + 10 i}{4 + 2 (- 2 i) + 2 i \cdot 2 + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{10 + 10 i}{4 - 4 i + 2 i \cdot 2 + 2 i (- 2 i)}$

Multiply $2$ by $2$ .

$\frac{10 + 10 i}{4 - 4 i + 4 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{10 + 10 i}{4 - 4 i + 4 i - 4 i i}$

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

$\frac{10 + 10 i}{4 - 4 i + 4 i - 4 (i^{1} i)}$

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

$\frac{10 + 10 i}{4 - 4 i + 4 i - 4 (i^{1} i^{1})}$

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

$\frac{10 + 10 i}{4 - 4 i + 4 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{10 + 10 i}{4 - 4 i + 4 i - 4 i^{2}}$

Add $- 4 i$ and $4 i$ .

$\frac{10 + 10 i}{4 + 0 - 4 i^{2}}$

Add $4$ and $0$ .

$\frac{10 + 10 i}{4 - 4 i^{2}}$

Simplify each term.

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

$\frac{10 + 10 i}{4 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{10 + 10 i}{4 + 4}$

Add $4$ and $4$ .

$\frac{10 + 10 i}{8}$

Step 3

Cancel the common factor of $10 + 10 i$ and $8$ .

Factor $2$ out of $10$ .

$\frac{2 (5) + 10 i}{8}$

Factor $2$ out of $10 i$ .

$\frac{2 (5) + 2 (5 i)}{8}$

Factor $2$ out of $2 (5) + 2 (5 i)$ .

$\frac{2 (5 + 5 i)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 (5 + 5 i)}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 (5 + 5 i)}{2 \cdot 4}$

Rewrite the expression.

$\frac{5 + 5 i}{4}$

Step 4

Split the fraction $\frac{5 + 5 i}{4}$ into two fractions.

$\frac{5}{4} + \frac{5 i}{4}$

Write in Standard Form (5i)/(2+2i)

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