Precalculus Examples

$(\frac{2}{3} + 24 i) (\frac{1}{6} + 12 i)$

Step 1

Set $(\frac{2}{3} + 24 i) (\frac{1}{6} + 12 i)$ equal to $0$ .

$(\frac{2}{3} + 24 i) (\frac{1}{6} + 12 i) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $(\frac{2}{3} + 24 i) (\frac{1}{6} + 12 i)$ .

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Step 2.1.1

Expand $(\frac{2}{3} + 24 i) (\frac{1}{6} + 12 i)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$\frac{2}{3} (\frac{1}{6} + 12 i) + 24 i (\frac{1}{6} + 12 i) = 0$

Step 2.1.1.2

Apply the distributive property.

$\frac{2}{3} \cdot \frac{1}{6} + \frac{2}{3} (12 i) + 24 i (\frac{1}{6} + 12 i) = 0$

Step 2.1.1.3

Apply the distributive property.

$\frac{2}{3} \cdot \frac{1}{6} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1

Factor $2$ out of $6$ .

$\frac{2}{3} \cdot \frac{1}{2 (3)} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.1.2

Cancel the common factor.

$\frac{2}{3} \cdot \frac{1}{2 \cdot 3} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.1.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{1}{3} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.2

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$\frac{1}{3 \cdot 3} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.3

Multiply $3$ by $3$ .

$\frac{1}{9} + \frac{2}{3} (12 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.4

Cancel the common factor of $3$ .

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Step 2.1.2.1.4.1

Factor $3$ out of $12 i$ .

$\frac{1}{9} + \frac{2}{3} (3 (4 i)) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.4.2

Cancel the common factor.

$\frac{1}{9} + \frac{2}{3} (3 (4 i)) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.4.3

Rewrite the expression.

$\frac{1}{9} + 2 (4 i) + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.5

Multiply $4$ by $2$ .

$\frac{1}{9} + 8 i + 24 i \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.6

Cancel the common factor of $6$ .

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Step 2.1.2.1.6.1

Factor $6$ out of $24 i$ .

$\frac{1}{9} + 8 i + 6 (4 i) \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.6.2

Cancel the common factor.

$\frac{1}{9} + 8 i + 6 (4 i) \frac{1}{6} + 24 i (12 i) = 0$

Step 2.1.2.1.6.3

Rewrite the expression.

$\frac{1}{9} + 8 i + 4 i + 24 i (12 i) = 0$

Step 2.1.2.1.7

Multiply $24 i (12 i)$ .

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Step 2.1.2.1.7.1

Multiply $12$ by $24$ .

$\frac{1}{9} + 8 i + 4 i + 288 i i = 0$

Step 2.1.2.1.7.2

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

$\frac{1}{9} + 8 i + 4 i + 288 (i^{1} i) = 0$

Step 2.1.2.1.7.3

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

$\frac{1}{9} + 8 i + 4 i + 288 (i^{1} i^{1}) = 0$

Step 2.1.2.1.7.4

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

$\frac{1}{9} + 8 i + 4 i + 288 i^{1 + 1} = 0$

Step 2.1.2.1.7.5

Add $1$ and $1$ .

$\frac{1}{9} + 8 i + 4 i + 288 i^{2} = 0$

Step 2.1.2.1.8

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

$\frac{1}{9} + 8 i + 4 i + 288 \cdot - 1 = 0$

Step 2.1.2.1.9

Multiply $288$ by $- 1$ .

$\frac{1}{9} + 8 i + 4 i - 288 = 0$

Step 2.1.2.2

To write $- 288$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{1}{9} - 288 \cdot \frac{9}{9} + 8 i + 4 i = 0$

Step 2.1.2.3

Combine $- 288$ and $\frac{9}{9}$ .

$\frac{1}{9} + \frac{- 288 \cdot 9}{9} + 8 i + 4 i = 0$

Step 2.1.2.4

Combine the numerators over the common denominator.

$\frac{1 - 288 \cdot 9}{9} + 8 i + 4 i = 0$

Step 2.1.2.5

Simplify the numerator.

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Step 2.1.2.5.1

Multiply $- 288$ by $9$ .

$\frac{1 - 2592}{9} + 8 i + 4 i = 0$

Step 2.1.2.5.2

Subtract $2592$ from $1$ .

$\frac{- 2591}{9} + 8 i + 4 i = 0$

Step 2.1.2.6

Add $8 i$ and $4 i$ .

$\frac{- 2591}{9} + 12 i = 0$

Step 2.1.3

Move the negative in front of the fraction.

$- \frac{2591}{9} + 12 i = 0$

Step 3

Since $- \frac{2591}{9} + 12 i \neq 0$ , there are no solutions.

No solution