Basic Math Examples

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

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

Step 1

Multiply the numerator and denominator of

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

$\frac{- 3 + 9 i}{3 + 7 i}$ by the conjugate of

3 + 7 i

$3 + 7 i$ to make the denominator real.

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

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

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(- 3 + 9 i) (3 - 7 i)}{(3 + 7 i) (3 - 7 i)}

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(- 3 + 9 i) (3 - 7 i)

$(- 3 + 9 i) (3 - 7 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{- 3 (3 - 7 i) + 9 i (3 - 7 i)}{(3 + 7 i) (3 - 7 i)}

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

Step 2.2.1.2

Apply the distributive property.

\frac{- 3 \cdot 3 - 3 (- 7 i) + 9 i (3 - 7 i)}{(3 + 7 i) (3 - 7 i)}

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

Step 2.2.1.3

Apply the distributive property.

\frac{- 3 \cdot 3 - 3 (- 7 i) + 9 i \cdot 3 + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}

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

\frac{- 3 \cdot 3 - 3 (- 7 i) + 9 i \cdot 3 + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}

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

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

- 3

$- 3$ by

3

$3$ .

\frac{- 9 - 3 (- 7 i) + 9 i \cdot 3 + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}

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

Step 2.2.2.1.2

Multiply

- 7

$- 7$ by

- 3

$- 3$ .

\frac{- 9 + 21 i + 9 i \cdot 3 + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 9 i \cdot 3 + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.3

Multiply

3

$3$ by

9

$9$ .

\frac{- 9 + 21 i + 27 i + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i + 9 i (- 7 i)}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.4

Multiply

9 i (- 7 i)

$9 i (- 7 i)$ .

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Step 2.2.2.1.4.1

Multiply

- 7

$- 7$ by

9

$9$ .

\frac{- 9 + 21 i + 27 i - 63 i i}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 i i}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.4.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 9 + 21 i + 27 i - 63 (i^{1} i)}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 (i^{1} i)}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.4.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 9 + 21 i + 27 i - 63 (i^{1} i^{1})}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 (i^{1} i^{1})}{(3 + 7 i) (3 - 7 i)}$

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{- 9 + 21 i + 27 i - 63 i^{1 + 1}}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 i^{1 + 1}}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.4.5

Add

1

$1$ and

1

$1$ .

\frac{- 9 + 21 i + 27 i - 63 i^{2}}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 i^{2}}{(3 + 7 i) (3 - 7 i)}$

\frac{- 9 + 21 i + 27 i - 63 i^{2}}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 i^{2}}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 9 + 21 i + 27 i - 63 \cdot - 1}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i - 63 \cdot - 1}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.1.6

Multiply

- 63

$- 63$ by

- 1

$- 1$ .

\frac{- 9 + 21 i + 27 i + 63}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i + 63}{(3 + 7 i) (3 - 7 i)}$

\frac{- 9 + 21 i + 27 i + 63}{(3 + 7 i) (3 - 7 i)}

$\frac{- 9 + 21 i + 27 i + 63}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.2

Add

- 9

$- 9$ and

63

$63$ .

\frac{54 + 21 i + 27 i}{(3 + 7 i) (3 - 7 i)}

$\frac{54 + 21 i + 27 i}{(3 + 7 i) (3 - 7 i)}$

Step 2.2.2.3

Add

21 i

$21 i$ and

27 i

$27 i$ .

\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}

$\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}$

\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}

$\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}$

\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}

$\frac{54 + 48 i}{(3 + 7 i) (3 - 7 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(3 + 7 i) (3 - 7 i)

$(3 + 7 i) (3 - 7 i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{54 + 48 i}{3 (3 - 7 i) + 7 i (3 - 7 i)}

$\frac{54 + 48 i}{3 (3 - 7 i) + 7 i (3 - 7 i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i (3 - 7 i)}

$\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i (3 - 7 i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}

$\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}$

\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}

$\frac{54 + 48 i}{3 \cdot 3 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

3

$3$ by

3

$3$ .

\frac{54 + 48 i}{9 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}

$\frac{54 + 48 i}{9 + 3 (- 7 i) + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2.2

Multiply

- 7

$- 7$ by

3

$3$ .

\frac{54 + 48 i}{9 - 21 i + 7 i \cdot 3 + 7 i (- 7 i)}

$\frac{54 + 48 i}{9 - 21 i + 7 i \cdot 3 + 7 i (- 7 i)}$

Step 2.3.2.3

Multiply

3

$3$ by

7

$7$ .

\frac{54 + 48 i}{9 - 21 i + 21 i + 7 i (- 7 i)}

$\frac{54 + 48 i}{9 - 21 i + 21 i + 7 i (- 7 i)}$

Step 2.3.2.4

Multiply

- 7

$- 7$ by

7

$7$ .

\frac{54 + 48 i}{9 - 21 i + 21 i - 49 i i}

$\frac{54 + 48 i}{9 - 21 i + 21 i - 49 i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{54 + 48 i}{9 - 21 i + 21 i - 49 (i^{1} i)}

$\frac{54 + 48 i}{9 - 21 i + 21 i - 49 (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{54 + 48 i}{9 - 21 i + 21 i - 49 (i^{1} i^{1})}

$\frac{54 + 48 i}{9 - 21 i + 21 i - 49 (i^{1} i^{1})}$

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{54 + 48 i}{9 - 21 i + 21 i - 49 i^{1 + 1}}

$\frac{54 + 48 i}{9 - 21 i + 21 i - 49 i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{54 + 48 i}{9 - 21 i + 21 i - 49 i^{2}}

$\frac{54 + 48 i}{9 - 21 i + 21 i - 49 i^{2}}$

Step 2.3.2.9

Add

- 21 i

$- 21 i$ and

21 i

$21 i$ .

\frac{54 + 48 i}{9 + 0 - 49 i^{2}}

$\frac{54 + 48 i}{9 + 0 - 49 i^{2}}$

Step 2.3.2.10

Add

9

$9$ and

0

$0$ .

\frac{54 + 48 i}{9 - 49 i^{2}}

$\frac{54 + 48 i}{9 - 49 i^{2}}$

\frac{54 + 48 i}{9 - 49 i^{2}}

$\frac{54 + 48 i}{9 - 49 i^{2}}$

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{54 + 48 i}{9 - 49 \cdot - 1}

$\frac{54 + 48 i}{9 - 49 \cdot - 1}$

Step 2.3.3.2

Multiply

- 49

$- 49$ by

- 1

$- 1$ .

\frac{54 + 48 i}{9 + 49}

$\frac{54 + 48 i}{9 + 49}$

\frac{54 + 48 i}{9 + 49}

$\frac{54 + 48 i}{9 + 49}$

Step 2.3.4

Add

9

$9$ and

49

$49$ .

\frac{54 + 48 i}{58}

$\frac{54 + 48 i}{58}$

\frac{54 + 48 i}{58}

$\frac{54 + 48 i}{58}$

\frac{54 + 48 i}{58}

$\frac{54 + 48 i}{58}$

Step 3

Cancel the common factor of

54 + 48 i

$54 + 48 i$ and

58

$58$ .

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Step 3.1

Factor

2

$2$ out of

54

$54$ .

\frac{2 (27) + 48 i}{58}

$\frac{2 (27) + 48 i}{58}$

Step 3.2

Factor

2

$2$ out of

48 i

$48 i$ .

\frac{2 (27) + 2 (24 i)}{58}

$\frac{2 (27) + 2 (24 i)}{58}$

Step 3.3

Factor

2

$2$ out of

2 (27) + 2 (24 i)

$2 (27) + 2 (24 i)$ .

\frac{2 (27 + 24 i)}{58}

$\frac{2 (27 + 24 i)}{58}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

2

$2$ out of

58

$58$ .

\frac{2 (27 + 24 i)}{2 \cdot 29}

$\frac{2 (27 + 24 i)}{2 \cdot 29}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (27 + 24 i)}{2 \cdot 29}$

Step 3.4.3

Rewrite the expression.

$\frac{27 + 24 i}{29}$

Step 4

Split the fraction

$\frac{27 + 24 i}{29}$ into two fractions.

$\frac{27}{29} + \frac{24 i}{29}$