Precalculus Examples

$(3 + 5 i) \cdot 2 \cdot (1.5 - 2.5 i)$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$(3 \cdot 2 + 5 i \cdot 2) \cdot (1.5 - 2.5 i)$

Step 1.2

Multiply.

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Step 1.2.1

Multiply $3$ by $2$ .

$(6 + 5 i \cdot 2) \cdot (1.5 - 2.5 i)$

Step 1.2.2

Multiply $2$ by $5$ .

$(6 + 10 i) \cdot (1.5 - 2.5 i)$

Step 2

Expand $(6 + 10 i) (1.5 - 2.5 i)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$6 (1.5 - 2.5 i) + 10 i (1.5 - 2.5 i)$

Step 2.2

Apply the distributive property.

$6 \cdot 1.5 + 6 (- 2.5 i) + 10 i (1.5 - 2.5 i)$

Step 2.3

Apply the distributive property.

$6 \cdot 1.5 + 6 (- 2.5 i) + 10 i \cdot 1.5 + 10 i (- 2.5 i)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $6$ by $1.5$ .

$9 + 6 (- 2.5 i) + 10 i \cdot 1.5 + 10 i (- 2.5 i)$

Step 3.1.2

Multiply $- 2.5$ by $6$ .

$9 - 15 i + 10 i \cdot 1.5 + 10 i (- 2.5 i)$

Step 3.1.3

Multiply $1.5$ by $10$ .

$9 - 15 i + 15 i + 10 i (- 2.5 i)$

Step 3.1.4

Multiply $10 i (- 2.5 i)$ .

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Step 3.1.4.1

Multiply $- 2.5$ by $10$ .

$9 - 15 i + 15 i - 25 i i$

Step 3.1.4.2

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

$9 - 15 i + 15 i - 25 (i^{1} i)$

Step 3.1.4.3

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

$9 - 15 i + 15 i - 25 (i^{1} i^{1})$

Step 3.1.4.4

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

$9 - 15 i + 15 i - 25 i^{1 + 1}$

Step 3.1.4.5

Add $1$ and $1$ .

$9 - 15 i + 15 i - 25 i^{2}$

Step 3.1.5

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

$9 - 15 i + 15 i - 25 \cdot - 1$

Step 3.1.6

Multiply $- 25$ by $- 1$ .

$9 - 15 i + 15 i + 25$

Step 3.2

Add $9$ and $25$ .

$34 - 15 i + 15 i$

Step 3.3

Add $- 15 i$ and $15 i$ .

$34 + 0 i$

Step 4

Multiply $0$ by $i$ .

$34 + 0$

Step 5

Add $34$ and $0$ .

$34$