Find the Complex Conjugate (9i)/(-2+7i)

$\frac{9 i}{- 2 + 7 i}$

Multiply the numerator and denominator of $\frac{9 i}{- 2 + 7 i}$ by the conjugate of $- 2 + 7 i$ to make the denominator real.

$\frac{9 i}{- 2 + 7 i} \cdot \frac{- 2 - 7 i}{- 2 - 7 i}$

Multiply.

Combine.

$\frac{9 i (- 2 - 7 i)}{(- 2 + 7 i) (- 2 - 7 i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{9 i \cdot - 2 + 9 i (- 7 i)}{(- 2 + 7 i) (- 2 - 7 i)}$

Multiply $- 2$ by $9$ .

$\frac{- 18 i + 9 i (- 7 i)}{(- 2 + 7 i) (- 2 - 7 i)}$

Multiply $9 i (- 7 i)$ .

Multiply $- 7$ by $9$ .

$\frac{- 18 i - 63 i i}{(- 2 + 7 i) (- 2 - 7 i)}$

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

$\frac{- 18 i - 63 (i^{1} i)}{(- 2 + 7 i) (- 2 - 7 i)}$

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

$\frac{- 18 i - 63 (i^{1} i^{1})}{(- 2 + 7 i) (- 2 - 7 i)}$

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

$\frac{- 18 i - 63 i^{1 + 1}}{(- 2 + 7 i) (- 2 - 7 i)}$

Add $1$ and $1$ .

$\frac{- 18 i - 63 i^{2}}{(- 2 + 7 i) (- 2 - 7 i)}$

Simplify each term.

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

$\frac{- 18 i - 63 \cdot - 1}{(- 2 + 7 i) (- 2 - 7 i)}$

Multiply $- 63$ by $- 1$ .

$\frac{- 18 i + 63}{(- 2 + 7 i) (- 2 - 7 i)}$

Reorder $- 18 i$ and $63$ .

$\frac{63 - 18 i}{(- 2 + 7 i) (- 2 - 7 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{63 - 18 i}{- 2 (- 2 - 7 i) + 7 i (- 2 - 7 i)}$

Apply the distributive property.

$\frac{63 - 18 i}{- 2 \cdot - 2 - 2 (- 7 i) + 7 i (- 2 - 7 i)}$

Apply the distributive property.

$\frac{63 - 18 i}{- 2 \cdot - 2 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)}$

Simplify.

Multiply $- 2$ by $- 2$ .

$\frac{63 - 18 i}{4 - 2 (- 7 i) + 7 i \cdot - 2 + 7 i (- 7 i)}$

Multiply $- 7$ by $- 2$ .

$\frac{63 - 18 i}{4 + 14 i + 7 i \cdot - 2 + 7 i (- 7 i)}$

Multiply $- 2$ by $7$ .

$\frac{63 - 18 i}{4 + 14 i - 14 i + 7 i (- 7 i)}$

Multiply $- 7$ by $7$ .

$\frac{63 - 18 i}{4 + 14 i - 14 i - 49 i i}$

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

$\frac{63 - 18 i}{4 + 14 i - 14 i - 49 (i^{1} i)}$

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

$\frac{63 - 18 i}{4 + 14 i - 14 i - 49 (i^{1} i^{1})}$

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

$\frac{63 - 18 i}{4 + 14 i - 14 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{63 - 18 i}{4 + 14 i - 14 i - 49 i^{2}}$

Subtract $14 i$ from $14 i$ .

$\frac{63 - 18 i}{4 + 0 - 49 i^{2}}$

Add $4$ and $0$ .

$\frac{63 - 18 i}{4 - 49 i^{2}}$

Simplify each term.

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

$\frac{63 - 18 i}{4 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{63 - 18 i}{4 + 49}$

Add $4$ and $49$ .

$\frac{63 - 18 i}{53}$

Split the fraction $\frac{63 - 18 i}{53}$ into two fractions.

$\frac{63}{53} + \frac{- 18 i}{53}$

Move the negative in front of the fraction.

$\frac{63}{53} - \frac{18 i}{53}$

To find the complex conjugate, negate the term with $i$ .

$\frac{63}{53} + \frac{18 i}{53}$

Find the Complex Conjugate (9i)/(-2+7i)

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