Simplify i/(-9+8i)

$\frac{i}{- 9 + 8 i}$

Step 1

Multiply the numerator and denominator of $\frac{i}{- 9 + 8 i}$ by the conjugate of $- 9 + 8 i$ to make the denominator real.

$\frac{i}{- 9 + 8 i} \cdot \frac{- 9 - 8 i}{- 9 - 8 i}$

Step 2

Multiply.

Combine.

$\frac{i (- 9 - 8 i)}{(- 9 + 8 i) (- 9 - 8 i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{i \cdot - 9 + i (- 8 i)}{(- 9 + 8 i) (- 9 - 8 i)}$

Move $- 9$ to the left of $i$ .

$\frac{- 9 \cdot i + i (- 8 i)}{(- 9 + 8 i) (- 9 - 8 i)}$

Multiply $i (- 8 i)$ .

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

$\frac{- 9 \cdot i - 8 (i^{1} i)}{(- 9 + 8 i) (- 9 - 8 i)}$

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

$\frac{- 9 \cdot i - 8 (i^{1} i^{1})}{(- 9 + 8 i) (- 9 - 8 i)}$

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

$\frac{- 9 \cdot i - 8 i^{1 + 1}}{(- 9 + 8 i) (- 9 - 8 i)}$

Add $1$ and $1$ .

$\frac{- 9 \cdot i - 8 i^{2}}{(- 9 + 8 i) (- 9 - 8 i)}$

Simplify each term.

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

$\frac{- 9 i - 8 \cdot - 1}{(- 9 + 8 i) (- 9 - 8 i)}$

Multiply $- 8$ by $- 1$ .

$\frac{- 9 i + 8}{(- 9 + 8 i) (- 9 - 8 i)}$

Reorder $- 9 i$ and $8$ .

$\frac{8 - 9 i}{(- 9 + 8 i) (- 9 - 8 i)}$

Simplify the denominator.

Expand $(- 9 + 8 i) (- 9 - 8 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 - 9 i}{- 9 (- 9 - 8 i) + 8 i (- 9 - 8 i)}$

Apply the distributive property.

$\frac{8 - 9 i}{- 9 \cdot - 9 - 9 (- 8 i) + 8 i (- 9 - 8 i)}$

Apply the distributive property.

$\frac{8 - 9 i}{- 9 \cdot - 9 - 9 (- 8 i) + 8 i \cdot - 9 + 8 i (- 8 i)}$

Simplify.

Multiply $- 9$ by $- 9$ .

$\frac{8 - 9 i}{81 - 9 (- 8 i) + 8 i \cdot - 9 + 8 i (- 8 i)}$

Multiply $- 8$ by $- 9$ .

$\frac{8 - 9 i}{81 + 72 i + 8 i \cdot - 9 + 8 i (- 8 i)}$

Multiply $- 9$ by $8$ .

$\frac{8 - 9 i}{81 + 72 i - 72 i + 8 i (- 8 i)}$

Multiply $- 8$ by $8$ .

$\frac{8 - 9 i}{81 + 72 i - 72 i - 64 i i}$

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

$\frac{8 - 9 i}{81 + 72 i - 72 i - 64 (i^{1} i)}$

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

$\frac{8 - 9 i}{81 + 72 i - 72 i - 64 (i^{1} i^{1})}$

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

$\frac{8 - 9 i}{81 + 72 i - 72 i - 64 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{8 - 9 i}{81 + 72 i - 72 i - 64 i^{2}}$

Subtract $72 i$ from $72 i$ .

$\frac{8 - 9 i}{81 + 0 - 64 i^{2}}$

Add $81$ and $0$ .

$\frac{8 - 9 i}{81 - 64 i^{2}}$

Simplify each term.

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

$\frac{8 - 9 i}{81 - 64 \cdot - 1}$

Multiply $- 64$ by $- 1$ .

$\frac{8 - 9 i}{81 + 64}$

Add $81$ and $64$ .

$\frac{8 - 9 i}{145}$

Step 3

Split the fraction $\frac{8 - 9 i}{145}$ into two fractions.

$\frac{8}{145} + \frac{- 9 i}{145}$

Step 4

Move the negative in front of the fraction.

$\frac{8}{145} - \frac{9 i}{145}$

Simplify i/(-9+8i)

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