Evaluate (6-7i)^-1

${(6 - 7 i)}^{- 1}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{6 - 7 i}$

Multiply the numerator and denominator of $\frac{1}{6 - 7 i}$ by the conjugate of $6 - 7 i$ to make the denominator real.

$\frac{1}{6 - 7 i} \cdot \frac{6 + 7 i}{6 + 7 i}$

Multiply.

Combine.

$\frac{1 (6 + 7 i)}{(6 - 7 i) (6 + 7 i)}$

Multiply $6 + 7 i$ by $1$ .

$\frac{6 + 7 i}{(6 - 7 i) (6 + 7 i)}$

Simplify the denominator.

Expand $(6 - 7 i) (6 + 7 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 + 7 i}{6 (6 + 7 i) - 7 i (6 + 7 i)}$

Apply the distributive property.

$\frac{6 + 7 i}{6 \cdot 6 + 6 (7 i) - 7 i (6 + 7 i)}$

Apply the distributive property.

$\frac{6 + 7 i}{6 \cdot 6 + 6 (7 i) - 7 i \cdot 6 - 7 i (7 i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{6 + 7 i}{36 + 6 (7 i) - 7 i \cdot 6 - 7 i (7 i)}$

Multiply $7$ by $6$ .

$\frac{6 + 7 i}{36 + 42 i - 7 i \cdot 6 - 7 i (7 i)}$

Multiply $6$ by $- 7$ .

$\frac{6 + 7 i}{36 + 42 i - 42 i - 7 i (7 i)}$

Multiply $7$ by $- 7$ .

$\frac{6 + 7 i}{36 + 42 i - 42 i - 49 i i}$

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

$\frac{6 + 7 i}{36 + 42 i - 42 i - 49 (i^{1} i)}$

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

$\frac{6 + 7 i}{36 + 42 i - 42 i - 49 (i^{1} i^{1})}$

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

$\frac{6 + 7 i}{36 + 42 i - 42 i - 49 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{6 + 7 i}{36 + 42 i - 42 i - 49 i^{2}}$

Subtract $42 i$ from $42 i$ .

$\frac{6 + 7 i}{36 + 0 - 49 i^{2}}$

Add $36$ and $0$ .

$\frac{6 + 7 i}{36 - 49 i^{2}}$

Simplify each term.

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

$\frac{6 + 7 i}{36 - 49 \cdot - 1}$

Multiply $- 49$ by $- 1$ .

$\frac{6 + 7 i}{36 + 49}$

Add $36$ and $49$ .

$\frac{6 + 7 i}{85}$

Split the fraction $\frac{6 + 7 i}{85}$ into two fractions.

$\frac{6}{85} + \frac{7 i}{85}$

Evaluate (6-7i)^-1

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