Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply.

Tap for more steps...

Step 3.1

Combine.

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

Step 3.2

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

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

Step 3.3

Simplify the denominator.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

Step 3.3.1.1

Apply the distributive property.

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

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Multiply $6$ by $6$ .

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

Step 3.3.2.2

Multiply $7$ by $6$ .

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

Step 3.3.2.3

Multiply $6$ by $- 7$ .

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

Step 3.3.2.4

Multiply $7$ by $- 7$ .

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

Step 3.3.2.5

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

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

Step 3.3.2.6

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

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

Step 3.3.2.7

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

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

Step 3.3.2.8

Add $1$ and $1$ .

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

Step 3.3.2.9

Subtract $42 i$ from $42 i$ .

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

Step 3.3.2.10

Add $36$ and $0$ .

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

Step 3.3.3

Simplify each term.

Tap for more steps...

Step 3.3.3.1

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

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

Step 3.3.3.2

Multiply $- 49$ by $- 1$ .

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

Step 3.3.4

Add $36$ and $49$ .

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

Step 4

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

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