Simplify (5+5i)/(5-2i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(5 + 5 i) (5 + 2 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$\frac{25 + 5 (2 i) + 5 i \cdot 5 + 5 i (2 i)}{(5 - 2 i) (5 + 2 i)}$

Multiply $2$ by $5$ .

$\frac{25 + 10 i + 5 i \cdot 5 + 5 i (2 i)}{(5 - 2 i) (5 + 2 i)}$

Multiply $5$ by $5$ .

$\frac{25 + 10 i + 25 i + 5 i (2 i)}{(5 - 2 i) (5 + 2 i)}$

Multiply $5 i (2 i)$ .

Multiply $2$ by $5$ .

$\frac{25 + 10 i + 25 i + 10 i i}{(5 - 2 i) (5 + 2 i)}$

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

$\frac{25 + 10 i + 25 i + 10 (i^{1} i)}{(5 - 2 i) (5 + 2 i)}$

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

$\frac{25 + 10 i + 25 i + 10 (i^{1} i^{1})}{(5 - 2 i) (5 + 2 i)}$

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

$\frac{25 + 10 i + 25 i + 10 i^{1 + 1}}{(5 - 2 i) (5 + 2 i)}$

Add $1$ and $1$ .

$\frac{25 + 10 i + 25 i + 10 i^{2}}{(5 - 2 i) (5 + 2 i)}$

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

$\frac{25 + 10 i + 25 i + 10 \cdot - 1}{(5 - 2 i) (5 + 2 i)}$

Multiply $10$ by $- 1$ .

$\frac{25 + 10 i + 25 i - 10}{(5 - 2 i) (5 + 2 i)}$

Subtract $10$ from $25$ .

$\frac{15 + 10 i + 25 i}{(5 - 2 i) (5 + 2 i)}$

Add $10 i$ and $25 i$ .

$\frac{15 + 35 i}{(5 - 2 i) (5 + 2 i)}$

Simplify the denominator.

Expand $(5 - 2 i) (5 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{15 + 35 i}{5 (5 + 2 i) - 2 i (5 + 2 i)}$

Apply the distributive property.

$\frac{15 + 35 i}{5 \cdot 5 + 5 (2 i) - 2 i (5 + 2 i)}$

Apply the distributive property.

$\frac{15 + 35 i}{5 \cdot 5 + 5 (2 i) - 2 i \cdot 5 - 2 i (2 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{15 + 35 i}{25 + 5 (2 i) - 2 i \cdot 5 - 2 i (2 i)}$

Multiply $2$ by $5$ .

$\frac{15 + 35 i}{25 + 10 i - 2 i \cdot 5 - 2 i (2 i)}$

Multiply $5$ by $- 2$ .

$\frac{15 + 35 i}{25 + 10 i - 10 i - 2 i (2 i)}$

Multiply $2$ by $- 2$ .

$\frac{15 + 35 i}{25 + 10 i - 10 i - 4 i i}$

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

$\frac{15 + 35 i}{25 + 10 i - 10 i - 4 (i^{1} i)}$

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

$\frac{15 + 35 i}{25 + 10 i - 10 i - 4 (i^{1} i^{1})}$

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

$\frac{15 + 35 i}{25 + 10 i - 10 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{15 + 35 i}{25 + 10 i - 10 i - 4 i^{2}}$

Subtract $10 i$ from $10 i$ .

$\frac{15 + 35 i}{25 + 0 - 4 i^{2}}$

Add $25$ and $0$ .

$\frac{15 + 35 i}{25 - 4 i^{2}}$

Simplify each term.

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

$\frac{15 + 35 i}{25 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{15 + 35 i}{25 + 4}$

Add $25$ and $4$ .

$\frac{15 + 35 i}{29}$

Step 3

Split the fraction $\frac{15 + 35 i}{29}$ into two fractions.

$\frac{15}{29} + \frac{35 i}{29}$

Simplify (5+5i)/(5-2i)

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