Simplify (5-7i)÷(2-5i)

$(5 - 7 i) \div (2 - 5 i)$

Rewrite the division as a fraction.

$\frac{5 - 7 i}{2 - 5 i}$

Multiply the numerator and denominator of $\frac{5 - 7 i}{2 - 5 i}$ by the conjugate of $2 - 5 i$ to make the denominator real.

$\frac{5 - 7 i}{2 - 5 i} \cdot \frac{2 + 5 i}{2 + 5 i}$

Multiply.

Combine.

$\frac{(5 - 7 i) (2 + 5 i)}{(2 - 5 i) (2 + 5 i)}$

Simplify the numerator.

Expand $(5 - 7 i) (2 + 5 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (2 + 5 i) - 7 i (2 + 5 i)}{(2 - 5 i) (2 + 5 i)}$

Apply the distributive property.

$\frac{5 \cdot 2 + 5 (5 i) - 7 i (2 + 5 i)}{(2 - 5 i) (2 + 5 i)}$

Apply the distributive property.

$\frac{5 \cdot 2 + 5 (5 i) - 7 i \cdot 2 - 7 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $2$ .

$\frac{10 + 5 (5 i) - 7 i \cdot 2 - 7 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $5$ by $5$ .

$\frac{10 + 25 i - 7 i \cdot 2 - 7 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $2$ by $- 7$ .

$\frac{10 + 25 i - 14 i - 7 i (5 i)}{(2 - 5 i) (2 + 5 i)}$

Multiply $- 7 i (5 i)$ .

Multiply $5$ by $- 7$ .

$\frac{10 + 25 i - 14 i - 35 i i}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{10 + 25 i - 14 i - 35 (i^{1} i)}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{10 + 25 i - 14 i - 35 (i^{1} i^{1})}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{10 + 25 i - 14 i - 35 i^{1 + 1}}{(2 - 5 i) (2 + 5 i)}$

Add $1$ and $1$ .

$\frac{10 + 25 i - 14 i - 35 i^{2}}{(2 - 5 i) (2 + 5 i)}$

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

$\frac{10 + 25 i - 14 i - 35 \cdot - 1}{(2 - 5 i) (2 + 5 i)}$

Multiply $- 35$ by $- 1$ .

$\frac{10 + 25 i - 14 i + 35}{(2 - 5 i) (2 + 5 i)}$

Add $10$ and $35$ .

$\frac{45 + 25 i - 14 i}{(2 - 5 i) (2 + 5 i)}$

Subtract $14 i$ from $25 i$ .

$\frac{45 + 11 i}{(2 - 5 i) (2 + 5 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{45 + 11 i}{2 (2 + 5 i) - 5 i (2 + 5 i)}$

Apply the distributive property.

$\frac{45 + 11 i}{2 \cdot 2 + 2 (5 i) - 5 i (2 + 5 i)}$

Apply the distributive property.

$\frac{45 + 11 i}{2 \cdot 2 + 2 (5 i) - 5 i \cdot 2 - 5 i (5 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{45 + 11 i}{4 + 2 (5 i) - 5 i \cdot 2 - 5 i (5 i)}$

Multiply $5$ by $2$ .

$\frac{45 + 11 i}{4 + 10 i - 5 i \cdot 2 - 5 i (5 i)}$

Multiply $2$ by $- 5$ .

$\frac{45 + 11 i}{4 + 10 i - 10 i - 5 i (5 i)}$

Multiply $5$ by $- 5$ .

$\frac{45 + 11 i}{4 + 10 i - 10 i - 25 i i}$

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

$\frac{45 + 11 i}{4 + 10 i - 10 i - 25 (i^{1} i)}$

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

$\frac{45 + 11 i}{4 + 10 i - 10 i - 25 (i^{1} i^{1})}$

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

$\frac{45 + 11 i}{4 + 10 i - 10 i - 25 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{45 + 11 i}{4 + 10 i - 10 i - 25 i^{2}}$

Subtract $10 i$ from $10 i$ .

$\frac{45 + 11 i}{4 + 0 - 25 i^{2}}$

Add $4$ and $0$ .

$\frac{45 + 11 i}{4 - 25 i^{2}}$

Simplify each term.

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

$\frac{45 + 11 i}{4 - 25 \cdot - 1}$

Multiply $- 25$ by $- 1$ .

$\frac{45 + 11 i}{4 + 25}$

Add $4$ and $25$ .

$\frac{45 + 11 i}{29}$

Split the fraction $\frac{45 + 11 i}{29}$ into two fractions.

$\frac{45}{29} + \frac{11 i}{29}$

Simplify (5-7i)÷(2-5i)

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