Algebra Examples

\frac{6 + 2 i}{4 - 7 i}

$\frac{6 + 2 i}{4 - 7 i}$

Step 1

Multiply the numerator and denominator of

\frac{6 + 2 i}{4 - 7 i}

$\frac{6 + 2 i}{4 - 7 i}$ by the conjugate of

4 - 7 i

$4 - 7 i$ to make the denominator real.

\frac{6 + 2 i}{4 - 7 i} \cdot \frac{4 + 7 i}{4 + 7 i}

$\frac{6 + 2 i}{4 - 7 i} \cdot \frac{4 + 7 i}{4 + 7 i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(6 + 2 i) (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{(6 + 2 i) (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(6 + 2 i) (4 + 7 i)

$(6 + 2 i) (4 + 7 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{6 (4 + 7 i) + 2 i (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{6 (4 + 7 i) + 2 i (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{6 \cdot 4 + 6 (7 i) + 2 i (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{6 \cdot 4 + 6 (7 i) + 2 i (4 + 7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{6 \cdot 4 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{6 \cdot 4 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}$

\frac{6 \cdot 4 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{6 \cdot 4 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}$

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

6

$6$ by

4

$4$ .

\frac{24 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 6 (7 i) + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.2

Multiply

7

$7$ by

6

$6$ .

\frac{24 + 42 i + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 2 i \cdot 4 + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.3

Multiply

4

$4$ by

2

$2$ .

\frac{24 + 42 i + 8 i + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 2 i (7 i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.4

Multiply

2 i (7 i)

$2 i (7 i)$ .

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Step 2.2.2.1.4.1

Multiply

7

$7$ by

2

$2$ .

\frac{24 + 42 i + 8 i + 14 i i}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 i i}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{24 + 42 i + 8 i + 14 (i^{1} i)}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 (i^{1} i)}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{24 + 42 i + 8 i + 14 (i^{1} i^{1})}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 (i^{1} i^{1})}{(4 - 7 i) (4 + 7 i)}$

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{24 + 42 i + 8 i + 14 i^{1 + 1}}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 i^{1 + 1}}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{24 + 42 i + 8 i + 14 i^{2}}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 i^{2}}{(4 - 7 i) (4 + 7 i)}$

\frac{24 + 42 i + 8 i + 14 i^{2}}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 i^{2}}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{24 + 42 i + 8 i + 14 \cdot - 1}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i + 14 \cdot - 1}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.1.6

Multiply

14

$14$ by

- 1

$- 1$ .

\frac{24 + 42 i + 8 i - 14}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i - 14}{(4 - 7 i) (4 + 7 i)}$

\frac{24 + 42 i + 8 i - 14}{(4 - 7 i) (4 + 7 i)}

$\frac{24 + 42 i + 8 i - 14}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.2

Subtract

14

$14$ from

24

$24$ .

\frac{10 + 42 i + 8 i}{(4 - 7 i) (4 + 7 i)}

$\frac{10 + 42 i + 8 i}{(4 - 7 i) (4 + 7 i)}$

Step 2.2.2.3

Add

42 i

$42 i$ and

8 i

$8 i$ .

\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}

$\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}$

\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}

$\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}$

\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}

$\frac{10 + 50 i}{(4 - 7 i) (4 + 7 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(4 - 7 i) (4 + 7 i)

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

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Step 2.3.1.1

Apply the distributive property.

\frac{10 + 50 i}{4 (4 + 7 i) - 7 i (4 + 7 i)}

$\frac{10 + 50 i}{4 (4 + 7 i) - 7 i (4 + 7 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i (4 + 7 i)}

$\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i (4 + 7 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}

$\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}$

\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}

$\frac{10 + 50 i}{4 \cdot 4 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

4

$4$ by

4

$4$ .

\frac{10 + 50 i}{16 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}

$\frac{10 + 50 i}{16 + 4 (7 i) - 7 i \cdot 4 - 7 i (7 i)}$

Step 2.3.2.2

Multiply

7

$7$ by

4

$4$ .

\frac{10 + 50 i}{16 + 28 i - 7 i \cdot 4 - 7 i (7 i)}

$\frac{10 + 50 i}{16 + 28 i - 7 i \cdot 4 - 7 i (7 i)}$

Step 2.3.2.3

Multiply

4

$4$ by

- 7

$- 7$ .

\frac{10 + 50 i}{16 + 28 i - 28 i - 7 i (7 i)}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 7 i (7 i)}$

Step 2.3.2.4

Multiply

7

$7$ by

- 7

$- 7$ .

\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i i}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{10 + 50 i}{16 + 28 i - 28 i - 49 (i^{1} i)}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 49 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{10 + 50 i}{16 + 28 i - 28 i - 49 (i^{1} i^{1})}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 49 (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

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

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

\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i^{1 + 1}}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i^{2}}

$\frac{10 + 50 i}{16 + 28 i - 28 i - 49 i^{2}}$

Step 2.3.2.9

Subtract

28 i

$28 i$ from

28 i

$28 i$ .

\frac{10 + 50 i}{16 + 0 - 49 i^{2}}

$\frac{10 + 50 i}{16 + 0 - 49 i^{2}}$

Step 2.3.2.10

Add

16

$16$ and

0

$0$ .

\frac{10 + 50 i}{16 - 49 i^{2}}

$\frac{10 + 50 i}{16 - 49 i^{2}}$

\frac{10 + 50 i}{16 - 49 i^{2}}

$\frac{10 + 50 i}{16 - 49 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{10 + 50 i}{16 - 49 \cdot - 1}

$\frac{10 + 50 i}{16 - 49 \cdot - 1}$

Step 2.3.3.2

Multiply

- 49

$- 49$ by

- 1

$- 1$ .

\frac{10 + 50 i}{16 + 49}

$\frac{10 + 50 i}{16 + 49}$

\frac{10 + 50 i}{16 + 49}

$\frac{10 + 50 i}{16 + 49}$

Step 2.3.4

Add

16

$16$ and

49

$49$ .

\frac{10 + 50 i}{65}

$\frac{10 + 50 i}{65}$

\frac{10 + 50 i}{65}

$\frac{10 + 50 i}{65}$

\frac{10 + 50 i}{65}

$\frac{10 + 50 i}{65}$

Step 3

Cancel the common factor of

10 + 50 i

$10 + 50 i$ and

65

$65$ .

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Step 3.1

Factor

5

$5$ out of

10

$10$ .

\frac{5 (2) + 50 i}{65}

$\frac{5 (2) + 50 i}{65}$

Step 3.2

Factor

5

$5$ out of

50 i

$50 i$ .

\frac{5 (2) + 5 (10 i)}{65}

$\frac{5 (2) + 5 (10 i)}{65}$

Step 3.3

Factor

5

$5$ out of

5 (2) + 5 (10 i)

$5 (2) + 5 (10 i)$ .

\frac{5 (2 + 10 i)}{65}

$\frac{5 (2 + 10 i)}{65}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

5

$5$ out of

65

$65$ .

\frac{5 (2 + 10 i)}{5 \cdot 13}

$\frac{5 (2 + 10 i)}{5 \cdot 13}$

Step 3.4.2

Cancel the common factor.

$\frac{5 (2 + 10 i)}{5 \cdot 13}$

Step 3.4.3

Rewrite the expression.

$\frac{2 + 10 i}{13}$

Step 4

Split the fraction

$\frac{2 + 10 i}{13}$ into two fractions.

$\frac{2}{13} + \frac{10 i}{13}$