Basic Math Examples

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

Step 1

Multiply the numerator and denominator of

$\frac{1 - 4 i}{3 - i}$ by the conjugate of

$3 - i$ to make the denominator real.

$\frac{1 - 4 i}{3 - i} \cdot \frac{3 + i}{3 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

$(1 - 4 i) (3 + i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply

$3$ by

$1$ .

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

Step 2.2.2.1.2

Multiply

$i$ by

$1$ .

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

Step 2.2.2.1.3

Multiply

$3$ by

$- 4$ .

$\frac{3 + i - 12 i - 4 i i}{(3 - i) (3 + i)}$

Step 2.2.2.1.4

Multiply

$- 4 i i$ .

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Step 2.2.2.1.4.1

Raise

$i$ to the power of

$1$ .

$\frac{3 + i - 12 i - 4 (i^{1} i)}{(3 - i) (3 + i)}$

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

$\frac{3 + i - 12 i - 4 (i^{1} i^{1})}{(3 - i) (3 + i)}$

Step 2.2.2.1.4.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 + i - 12 i - 4 i^{1 + 1}}{(3 - i) (3 + i)}$

Step 2.2.2.1.4.4

Add

$1$ and

$1$ .

$\frac{3 + i - 12 i - 4 i^{2}}{(3 - i) (3 + i)}$

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{3 + i - 12 i - 4 \cdot - 1}{(3 - i) (3 + i)}$

Step 2.2.2.1.6

Multiply

$- 4$ by

$- 1$ .

$\frac{3 + i - 12 i + 4}{(3 - i) (3 + i)}$

Step 2.2.2.2

Add

$3$ and

$4$ .

$\frac{7 + i - 12 i}{(3 - i) (3 + i)}$

Step 2.2.2.3

Subtract

$12 i$ from

$i$ .

$\frac{7 - 11 i}{(3 - i) (3 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{7 - 11 i}{3 (3 + i) - i (3 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{7 - 11 i}{3 \cdot 3 + 3 i - i (3 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{7 - 11 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$3$ by

$3$ .

$\frac{7 - 11 i}{9 + 3 i - i \cdot 3 - i i}$

Step 2.3.2.2

Multiply

$3$ by

$- 1$ .

$\frac{7 - 11 i}{9 + 3 i - 3 i - i i}$

Step 2.3.2.3

Raise

$i$ to the power of

$1$ .

$\frac{7 - 11 i}{9 + 3 i - 3 i - (i^{1} i)}$

Step 2.3.2.4

Raise

$i$ to the power of

$1$ .

$\frac{7 - 11 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

Step 2.3.2.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{7 - 11 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Step 2.3.2.6

Add

$1$ and

$1$ .

$\frac{7 - 11 i}{9 + 3 i - 3 i - i^{2}}$

Step 2.3.2.7

Subtract

$3 i$ from

$3 i$ .

$\frac{7 - 11 i}{9 + 0 - i^{2}}$

Step 2.3.2.8

Add

$9$ and

$0$ .

$\frac{7 - 11 i}{9 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{7 - 11 i}{9 - - 1}$

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

$\frac{7 - 11 i}{9 + 1}$

Step 2.3.4

Add

$9$ and

$1$ .

$\frac{7 - 11 i}{10}$

Step 3

Split the fraction

$\frac{7 - 11 i}{10}$ into two fractions.

$\frac{7}{10} + \frac{- 11 i}{10}$

Step 4

Move the negative in front of the fraction.

$\frac{7}{10} - \frac{11 i}{10}$