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

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $5$ .

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

Multiply $- 2$ by $2$ .

$\frac{10 - 4 i + 5 i \cdot 5 + 5 i (- 2 i)}{(5 + 2 i) (5 - 2 i)}$

Multiply $5$ by $5$ .

$\frac{10 - 4 i + 25 i + 5 i (- 2 i)}{(5 + 2 i) (5 - 2 i)}$

Multiply $5 i (- 2 i)$ .

Multiply $- 2$ by $5$ .

$\frac{10 - 4 i + 25 i - 10 i i}{(5 + 2 i) (5 - 2 i)}$

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

$\frac{10 - 4 i + 25 i - 10 (i^{1} i)}{(5 + 2 i) (5 - 2 i)}$

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

$\frac{10 - 4 i + 25 i - 10 (i^{1} i^{1})}{(5 + 2 i) (5 - 2 i)}$

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

$\frac{10 - 4 i + 25 i - 10 i^{1 + 1}}{(5 + 2 i) (5 - 2 i)}$

Add $1$ and $1$ .

$\frac{10 - 4 i + 25 i - 10 i^{2}}{(5 + 2 i) (5 - 2 i)}$

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

$\frac{10 - 4 i + 25 i - 10 \cdot - 1}{(5 + 2 i) (5 - 2 i)}$

Multiply $- 10$ by $- 1$ .

$\frac{10 - 4 i + 25 i + 10}{(5 + 2 i) (5 - 2 i)}$

Add $10$ and $10$ .

$\frac{20 - 4 i + 25 i}{(5 + 2 i) (5 - 2 i)}$

Add $- 4 i$ and $25 i$ .

$\frac{20 + 21 i}{(5 + 2 i) (5 - 2 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{20 + 21 i}{5 (5 - 2 i) + 2 i (5 - 2 i)}$

Apply the distributive property.

$\frac{20 + 21 i}{5 \cdot 5 + 5 (- 2 i) + 2 i (5 - 2 i)}$

Apply the distributive property.

$\frac{20 + 21 i}{5 \cdot 5 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{20 + 21 i}{25 + 5 (- 2 i) + 2 i \cdot 5 + 2 i (- 2 i)}$

Multiply $- 2$ by $5$ .

$\frac{20 + 21 i}{25 - 10 i + 2 i \cdot 5 + 2 i (- 2 i)}$

Multiply $5$ by $2$ .

$\frac{20 + 21 i}{25 - 10 i + 10 i + 2 i (- 2 i)}$

Multiply $- 2$ by $2$ .

$\frac{20 + 21 i}{25 - 10 i + 10 i - 4 i i}$

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

$\frac{20 + 21 i}{25 - 10 i + 10 i - 4 (i^{1} i)}$

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

$\frac{20 + 21 i}{25 - 10 i + 10 i - 4 (i^{1} i^{1})}$

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

$\frac{20 + 21 i}{25 - 10 i + 10 i - 4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{20 + 21 i}{25 - 10 i + 10 i - 4 i^{2}}$

Add $- 10 i$ and $10 i$ .

$\frac{20 + 21 i}{25 + 0 - 4 i^{2}}$

Add $25$ and $0$ .

$\frac{20 + 21 i}{25 - 4 i^{2}}$

Simplify each term.

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

$\frac{20 + 21 i}{25 - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$\frac{20 + 21 i}{25 + 4}$

Add $25$ and $4$ .

$\frac{20 + 21 i}{29}$

Step 3

Split the fraction $\frac{20 + 21 i}{29}$ into two fractions.

$\frac{20}{29} + \frac{21 i}{29}$

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

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