Basic Math Examples

$\frac{9 + \sqrt{2} i}{4 - 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{9 + \sqrt{2} i}{4 - 3 i}$ by the conjugate of $4 - 3 i$ to make the denominator real.

$\frac{9 + \sqrt{2} i}{4 - 3 i} \cdot \frac{4 + 3 i}{4 + 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(9 + \sqrt{2} i) (4 + 3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand $(9 + \sqrt{2} i) (4 + 3 i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$\frac{9 (4 + 3 i) + \sqrt{2} i (4 + 3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.1.2

Apply the distributive property.

$\frac{9 \cdot 4 + 9 (3 i) + \sqrt{2} i (4 + 3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{9 \cdot 4 + 9 (3 i) + \sqrt{2} i \cdot 4 + \sqrt{2} i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $9$ by $4$ .

$\frac{36 + 9 (3 i) + \sqrt{2} i \cdot 4 + \sqrt{2} i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.2

Multiply $3$ by $9$ .

$\frac{36 + 27 i + \sqrt{2} i \cdot 4 + \sqrt{2} i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.3

Move $4$ to the left of $\sqrt{2} i$ .

$\frac{36 + 27 i + 4 \cdot (\sqrt{2} i) + \sqrt{2} i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.4

Multiply $\sqrt{2} i (3 i)$ .

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Step 2.2.2.4.1

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

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} (3 (i^{1} i))}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.4.2

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

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} (3 (i^{1} i^{1}))}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.4.3

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

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} (3 i^{1 + 1})}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.4.4

Add $1$ and $1$ .

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} (3 i^{2})}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.5

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

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} (3 \cdot - 1)}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.6

Multiply $3$ by $- 1$ .

$\frac{36 + 27 i + 4 \sqrt{2} i + \sqrt{2} \cdot - 3}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.2.7

Move $- 3$ to the left of $\sqrt{2}$ .

$\frac{36 + 27 i + 4 \sqrt{2} i - 3 \sqrt{2}}{(4 - 3 i) (4 + 3 i)}$

Step 2.2.3

Reorder $36 + 27 i + 4 \sqrt{2} i$ and $- 3 \sqrt{2}$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{(4 - 3 i) (4 + 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{4 (4 + 3 i) - 3 i (4 + 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{4 \cdot 4 + 4 (3 i) - 3 i (4 + 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{4 \cdot 4 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $4$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2.2

Multiply $3$ by $4$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 3 i \cdot 4 - 3 i (3 i)}$

Step 2.3.2.3

Multiply $4$ by $- 3$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 3 i (3 i)}$

Step 2.3.2.4

Multiply $3$ by $- 3$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 9 i i}$

Step 2.3.2.5

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

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 12 i - 12 i - 9 i^{2}}$

Step 2.3.2.9

Subtract $12 i$ from $12 i$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $16$ and $0$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{16 + 9}$

Step 2.3.4

Add $16$ and $9$ .

$\frac{- 3 \sqrt{2} + 36 + 27 i + 4 \sqrt{2} i}{25}$

Step 3

Factor $- 1$ out of $- 3 \sqrt{2}$ .

$\frac{- (3 \sqrt{2}) + 36 + 27 i + 4 \sqrt{2} i}{25}$

Step 4

Rewrite $36$ as $- 1 (- 36)$ .

$\frac{- (3 \sqrt{2}) - 1 (- 36) + 27 i + 4 \sqrt{2} i}{25}$

Step 5

Factor $- 1$ out of $- (3 \sqrt{2}) - 1 (- 36)$ .

$\frac{- (3 \sqrt{2} - 36) + 27 i + 4 \sqrt{2} i}{25}$

Step 6

Factor $- 1$ out of $27 i$ .

$\frac{- (3 \sqrt{2} - 36) - (- 27 i) + 4 \sqrt{2} i}{25}$

Step 7

Factor $- 1$ out of $- (3 \sqrt{2} - 36) - (- 27 i)$ .

$\frac{- (3 \sqrt{2} - 36 - 27 i) + 4 \sqrt{2} i}{25}$

Step 8

Factor $- 1$ out of $4 \sqrt{2} i$ .

$\frac{- (3 \sqrt{2} - 36 - 27 i) - (- 4 \sqrt{2} i)}{25}$

Step 9

Factor $- 1$ out of $- (3 \sqrt{2} - 36 - 27 i) - (- 4 \sqrt{2} i)$ .

$\frac{- (3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i)}{25}$

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- (3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i)$ as $- 1 (3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i)$ .

$\frac{- 1 (3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i)}{25}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i}{25}$

Step 11

Split the fraction $\frac{3 \sqrt{2} - 36 - 27 i - 4 \sqrt{2} i}{25}$ into two fractions.

$- (\frac{3 \sqrt{2} - 36 - 27 i}{25} + \frac{- 4 \sqrt{2} i}{25})$

Step 12

Split the fraction $\frac{3 \sqrt{2} - 36 - 27 i}{25}$ into two fractions.

$- (\frac{3 \sqrt{2} - 36}{25} + \frac{- 27 i}{25} + \frac{- 4 \sqrt{2} i}{25})$

Step 13

Split the fraction $\frac{3 \sqrt{2} - 36}{25}$ into two fractions.

$- (\frac{3 \sqrt{2}}{25} + \frac{- 36}{25} + \frac{- 27 i}{25} + \frac{- 4 \sqrt{2} i}{25})$

Step 14

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{2}}{25} - \frac{36}{25} + \frac{- 27 i}{25} + \frac{- 4 \sqrt{2} i}{25})$

Step 15

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{2}}{25} - \frac{36}{25} - \frac{27 i}{25} + \frac{- 4 \sqrt{2} i}{25})$

Step 16

Move the negative in front of the fraction.

$- (\frac{3 \sqrt{2}}{25} - \frac{36}{25} - \frac{27 i}{25} - \frac{4 \sqrt{2} i}{25})$