Algebra Examples

${(\frac{3 - 4 i}{5})}^{- 1}$

Step 1

Split the fraction $\frac{3 - 4 i}{5}$ into two fractions.

${(\frac{3}{5} + \frac{- 4 i}{5})}^{- 1}$

Step 2

Move the negative in front of the fraction.

${(\frac{3}{5} - \frac{4 i}{5})}^{- 1}$

Step 3

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

$\frac{1}{\frac{3}{5} - \frac{4 i}{5}}$

Step 4

Combine the numerators over the common denominator.

$\frac{1}{\frac{3 - 4 i}{5}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{5}{3 - 4 i}$

Step 6

Multiply $\frac{5}{3 - 4 i}$ by $1$ .

$\frac{5}{3 - 4 i}$

Step 7

Multiply the numerator and denominator of $\frac{5}{3 - 4 i}$ by the conjugate of $3 - 4 i$ to make the denominator real.

$\frac{5}{3 - 4 i} \cdot \frac{3 + 4 i}{3 + 4 i}$

Step 8

Multiply.

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Step 8.1

Combine.

$\frac{5 (3 + 4 i)}{(3 - 4 i) (3 + 4 i)}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (4 i)}{(3 - 4 i) (3 + 4 i)}$

Step 8.2.2

Multiply $5$ by $3$ .

$\frac{15 + 5 (4 i)}{(3 - 4 i) (3 + 4 i)}$

Step 8.2.3

Multiply $4$ by $5$ .

$\frac{15 + 20 i}{(3 - 4 i) (3 + 4 i)}$

Step 8.3

Simplify the denominator.

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Step 8.3.1

Expand $(3 - 4 i) (3 + 4 i)$ using the FOIL Method.

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Step 8.3.1.1

Apply the distributive property.

$\frac{15 + 20 i}{3 (3 + 4 i) - 4 i (3 + 4 i)}$

Step 8.3.1.2

Apply the distributive property.

$\frac{15 + 20 i}{3 \cdot 3 + 3 (4 i) - 4 i (3 + 4 i)}$

Step 8.3.1.3

Apply the distributive property.

$\frac{15 + 20 i}{3 \cdot 3 + 3 (4 i) - 4 i \cdot 3 - 4 i (4 i)}$

Step 8.3.2

Simplify.

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Step 8.3.2.1

Multiply $3$ by $3$ .

$\frac{15 + 20 i}{9 + 3 (4 i) - 4 i \cdot 3 - 4 i (4 i)}$

Step 8.3.2.2

Multiply $4$ by $3$ .

$\frac{15 + 20 i}{9 + 12 i - 4 i \cdot 3 - 4 i (4 i)}$

Step 8.3.2.3

Multiply $3$ by $- 4$ .

$\frac{15 + 20 i}{9 + 12 i - 12 i - 4 i (4 i)}$

Step 8.3.2.4

Multiply $4$ by $- 4$ .

$\frac{15 + 20 i}{9 + 12 i - 12 i - 16 i i}$

Step 8.3.2.5

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

$\frac{15 + 20 i}{9 + 12 i - 12 i - 16 (i^{1} i)}$

Step 8.3.2.6

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

$\frac{15 + 20 i}{9 + 12 i - 12 i - 16 (i^{1} i^{1})}$

Step 8.3.2.7

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

$\frac{15 + 20 i}{9 + 12 i - 12 i - 16 i^{1 + 1}}$

Step 8.3.2.8

Add $1$ and $1$ .

$\frac{15 + 20 i}{9 + 12 i - 12 i - 16 i^{2}}$

Step 8.3.2.9

Subtract $12 i$ from $12 i$ .

$\frac{15 + 20 i}{9 + 0 - 16 i^{2}}$

Step 8.3.2.10

Add $9$ and $0$ .

$\frac{15 + 20 i}{9 - 16 i^{2}}$

Step 8.3.3

Simplify each term.

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Step 8.3.3.1

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

$\frac{15 + 20 i}{9 - 16 \cdot - 1}$

Step 8.3.3.2

Multiply $- 16$ by $- 1$ .

$\frac{15 + 20 i}{9 + 16}$

Step 8.3.4

Add $9$ and $16$ .

$\frac{15 + 20 i}{25}$

Step 9

Cancel the common factor of $15 + 20 i$ and $25$ .

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Step 9.1

Factor $5$ out of $15$ .

$\frac{5 (3) + 20 i}{25}$

Step 9.2

Factor $5$ out of $20 i$ .

$\frac{5 (3) + 5 (4 i)}{25}$

Step 9.3

Factor $5$ out of $5 (3) + 5 (4 i)$ .

$\frac{5 (3 + 4 i)}{25}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $5$ out of $25$ .

$\frac{5 (3 + 4 i)}{5 \cdot 5}$

Step 9.4.2

Cancel the common factor.

$\frac{5 (3 + 4 i)}{5 \cdot 5}$

Step 9.4.3

Rewrite the expression.

$\frac{3 + 4 i}{5}$

Step 10

Split the fraction $\frac{3 + 4 i}{5}$ into two fractions.

$\frac{3}{5} + \frac{4 i}{5}$