Basic Math Examples

$\frac{2 i (6 - 4 i)}{3 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{2 i (6 - 4 i)}{3 + 3 i}$ by the conjugate of $3 + 3 i$ to make the denominator real.

$\frac{2 i (6 - 4 i)}{3 + 3 i} \cdot \frac{3 - 3 i}{3 - 3 i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{2 i (6 - 4 i) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{(2 i \cdot 6 + 2 i (- 4 i)) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.2

Multiply $6$ by $2$ .

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

Step 2.2.3

Multiply $2 i (- 4 i)$ .

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Step 2.2.3.1

Multiply $- 4$ by $2$ .

$\frac{(12 i - 8 i i) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.3.2

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

$\frac{(12 i - 8 (i^{1} i)) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.3.3

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

$\frac{(12 i - 8 (i^{1} i^{1})) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.3.4

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

$\frac{(12 i - 8 i^{1 + 1}) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.3.5

Add $1$ and $1$ .

$\frac{(12 i - 8 i^{2}) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

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

$\frac{(12 i - 8 \cdot - 1) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.4.2

Multiply $- 8$ by $- 1$ .

$\frac{(12 i + 8) (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.5

Expand $(12 i + 8) (3 - 3 i)$ using the FOIL Method.

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Step 2.2.5.1

Apply the distributive property.

$\frac{12 i (3 - 3 i) + 8 (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.5.2

Apply the distributive property.

$\frac{12 i \cdot 3 + 12 i (- 3 i) + 8 (3 - 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.5.3

Apply the distributive property.

$\frac{12 i \cdot 3 + 12 i (- 3 i) + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6

Simplify and combine like terms.

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Step 2.2.6.1

Simplify each term.

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Step 2.2.6.1.1

Multiply $3$ by $12$ .

$\frac{36 i + 12 i (- 3 i) + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.2

Multiply $12 i (- 3 i)$ .

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Step 2.2.6.1.2.1

Multiply $- 3$ by $12$ .

$\frac{36 i - 36 i i + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.2.2

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

$\frac{36 i - 36 (i^{1} i) + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.2.3

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

$\frac{36 i - 36 (i^{1} i^{1}) + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.2.4

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

$\frac{36 i - 36 i^{1 + 1} + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.2.5

Add $1$ and $1$ .

$\frac{36 i - 36 i^{2} + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.3

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

$\frac{36 i - 36 \cdot - 1 + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.4

Multiply $- 36$ by $- 1$ .

$\frac{36 i + 36 + 8 \cdot 3 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.5

Multiply $8$ by $3$ .

$\frac{36 i + 36 + 24 + 8 (- 3 i)}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.1.6

Multiply $- 3$ by $8$ .

$\frac{36 i + 36 + 24 - 24 i}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.2

Subtract $24 i$ from $36 i$ .

$\frac{36 + 24 + 12 i}{(3 + 3 i) (3 - 3 i)}$

Step 2.2.6.3

Add $36$ and $24$ .

$\frac{60 + 12 i}{(3 + 3 i) (3 - 3 i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the distributive property.

$\frac{60 + 12 i}{3 (3 - 3 i) + 3 i (3 - 3 i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{60 + 12 i}{3 \cdot 3 + 3 (- 3 i) + 3 i (3 - 3 i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{60 + 12 i}{3 \cdot 3 + 3 (- 3 i) + 3 i \cdot 3 + 3 i (- 3 i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $3$ by $3$ .

$\frac{60 + 12 i}{9 + 3 (- 3 i) + 3 i \cdot 3 + 3 i (- 3 i)}$

Step 2.3.2.2

Multiply $- 3$ by $3$ .

$\frac{60 + 12 i}{9 - 9 i + 3 i \cdot 3 + 3 i (- 3 i)}$

Step 2.3.2.3

Multiply $3$ by $3$ .

$\frac{60 + 12 i}{9 - 9 i + 9 i + 3 i (- 3 i)}$

Step 2.3.2.4

Multiply $- 3$ by $3$ .

$\frac{60 + 12 i}{9 - 9 i + 9 i - 9 i i}$

Step 2.3.2.5

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

$\frac{60 + 12 i}{9 - 9 i + 9 i - 9 (i^{1} i)}$

Step 2.3.2.6

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

$\frac{60 + 12 i}{9 - 9 i + 9 i - 9 (i^{1} i^{1})}$

Step 2.3.2.7

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

$\frac{60 + 12 i}{9 - 9 i + 9 i - 9 i^{1 + 1}}$

Step 2.3.2.8

Add $1$ and $1$ .

$\frac{60 + 12 i}{9 - 9 i + 9 i - 9 i^{2}}$

Step 2.3.2.9

Add $- 9 i$ and $9 i$ .

$\frac{60 + 12 i}{9 + 0 - 9 i^{2}}$

Step 2.3.2.10

Add $9$ and $0$ .

$\frac{60 + 12 i}{9 - 9 i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

$\frac{60 + 12 i}{9 - 9 \cdot - 1}$

Step 2.3.3.2

Multiply $- 9$ by $- 1$ .

$\frac{60 + 12 i}{9 + 9}$

Step 2.3.4

Add $9$ and $9$ .

$\frac{60 + 12 i}{18}$

Step 3

Cancel the common factor of $60 + 12 i$ and $18$ .

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Step 3.1

Factor $6$ out of $60$ .

$\frac{6 (10) + 12 i}{18}$

Step 3.2

Factor $6$ out of $12 i$ .

$\frac{6 (10) + 6 (2 i)}{18}$

Step 3.3

Factor $6$ out of $6 (10) + 6 (2 i)$ .

$\frac{6 (10 + 2 i)}{18}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $6$ out of $18$ .

$\frac{6 (10 + 2 i)}{6 \cdot 3}$

Step 3.4.2

Cancel the common factor.

$\frac{6 (10 + 2 i)}{6 \cdot 3}$

Step 3.4.3

Rewrite the expression.

$\frac{10 + 2 i}{3}$

Step 4

Split the fraction $\frac{10 + 2 i}{3}$ into two fractions.

$\frac{10}{3} + \frac{2 i}{3}$