Basic Math Examples

\frac{4 + 4 i}{7 - 2 i}

$\frac{4 + 4 i}{7 - 2 i}$

Step 1

Multiply the numerator and denominator of

\frac{4 + 4 i}{7 - 2 i}

$\frac{4 + 4 i}{7 - 2 i}$ by the conjugate of

7 - 2 i

$7 - 2 i$ to make the denominator real.

\frac{4 + 4 i}{7 - 2 i} \cdot \frac{7 + 2 i}{7 + 2 i}

$\frac{4 + 4 i}{7 - 2 i} \cdot \frac{7 + 2 i}{7 + 2 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(4 + 4 i) (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{(4 + 4 i) (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(4 + 4 i) (7 + 2 i)

$(4 + 4 i) (7 + 2 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{4 (7 + 2 i) + 4 i (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{4 (7 + 2 i) + 4 i (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{4 \cdot 7 + 4 (2 i) + 4 i (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{4 \cdot 7 + 4 (2 i) + 4 i (7 + 2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{4 \cdot 7 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{4 \cdot 7 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}$

\frac{4 \cdot 7 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{4 \cdot 7 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}$

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

4

$4$ by

7

$7$ .

\frac{28 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 4 (2 i) + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.2

Multiply

2

$2$ by

4

$4$ .

\frac{28 + 8 i + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 4 i \cdot 7 + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.3

Multiply

7

$7$ by

4

$4$ .

\frac{28 + 8 i + 28 i + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 4 i (2 i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.4

Multiply

4 i (2 i)

$4 i (2 i)$ .

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Step 2.2.2.1.4.1

Multiply

2

$2$ by

4

$4$ .

\frac{28 + 8 i + 28 i + 8 i i}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 i i}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{28 + 8 i + 28 i + 8 (i^{1} i)}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 (i^{1} i)}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{28 + 8 i + 28 i + 8 (i^{1} i^{1})}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 (i^{1} i^{1})}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{28 + 8 i + 28 i + 8 i^{1 + 1}}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 i^{1 + 1}}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{28 + 8 i + 28 i + 8 i^{2}}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 i^{2}}{(7 - 2 i) (7 + 2 i)}$

\frac{28 + 8 i + 28 i + 8 i^{2}}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 i^{2}}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{28 + 8 i + 28 i + 8 \cdot - 1}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i + 8 \cdot - 1}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.1.6

Multiply

8

$8$ by

- 1

$- 1$ .

\frac{28 + 8 i + 28 i - 8}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i - 8}{(7 - 2 i) (7 + 2 i)}$

\frac{28 + 8 i + 28 i - 8}{(7 - 2 i) (7 + 2 i)}

$\frac{28 + 8 i + 28 i - 8}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.2

Subtract

8

$8$ from

28

$28$ .

\frac{20 + 8 i + 28 i}{(7 - 2 i) (7 + 2 i)}

$\frac{20 + 8 i + 28 i}{(7 - 2 i) (7 + 2 i)}$

Step 2.2.2.3

Add

8 i

$8 i$ and

28 i

$28 i$ .

\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}

$\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}$

\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}

$\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}$

\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}

$\frac{20 + 36 i}{(7 - 2 i) (7 + 2 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(7 - 2 i) (7 + 2 i)

$(7 - 2 i) (7 + 2 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{20 + 36 i}{7 (7 + 2 i) - 2 i (7 + 2 i)}

$\frac{20 + 36 i}{7 (7 + 2 i) - 2 i (7 + 2 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{20 + 36 i}{7 \cdot 7 + 7 (2 i) - 2 i (7 + 2 i)}

$\frac{20 + 36 i}{7 \cdot 7 + 7 (2 i) - 2 i (7 + 2 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{20 + 36 i}{7 \cdot 7 + 7 (2 i) - 2 i \cdot 7 - 2 i (2 i)}

$\frac{20 + 36 i}{7 \cdot 7 + 7 (2 i) - 2 i \cdot 7 - 2 i (2 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$7$ by

$7$ .

$\frac{20 + 36 i}{49 + 7 (2 i) - 2 i \cdot 7 - 2 i (2 i)}$

Step 2.3.2.2

Multiply

$2$ by

$7$ .

$\frac{20 + 36 i}{49 + 14 i - 2 i \cdot 7 - 2 i (2 i)}$

Step 2.3.2.3

Multiply

$7$ by

$- 2$ .

$\frac{20 + 36 i}{49 + 14 i - 14 i - 2 i (2 i)}$

Step 2.3.2.4

Multiply

$2$ by

$- 2$ .

$\frac{20 + 36 i}{49 + 14 i - 14 i - 4 i i}$

Step 2.3.2.5

Raise

$i$ to the power of

$1$ .

$\frac{20 + 36 i}{49 + 14 i - 14 i - 4 (i^{1} i)}$

Step 2.3.2.6

Raise

$i$ to the power of

$1$ .

$\frac{20 + 36 i}{49 + 14 i - 14 i - 4 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

$\frac{20 + 36 i}{49 + 14 i - 14 i - 4 i^{1 + 1}}$

Step 2.3.2.8

Add

$1$ and

$1$ .

$\frac{20 + 36 i}{49 + 14 i - 14 i - 4 i^{2}}$

Step 2.3.2.9

Subtract

$14 i$ from

$14 i$ .

$\frac{20 + 36 i}{49 + 0 - 4 i^{2}}$

Step 2.3.2.10

Add

$49$ and

$0$ .

$\frac{20 + 36 i}{49 - 4 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{20 + 36 i}{49 - 4 \cdot - 1}$

Step 2.3.3.2

Multiply

$- 4$ by

$- 1$ .

$\frac{20 + 36 i}{49 + 4}$

Step 2.3.4

Add

$49$ and

$4$ .

$\frac{20 + 36 i}{53}$

Step 3

Split the fraction

$\frac{20 + 36 i}{53}$ into two fractions.

$\frac{20}{53} + \frac{36 i}{53}$