Basic Math Examples

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

Step 1

Combine using the product rule for radicals.

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

Step 2

Expand $(3 + 2 i) (3 - 2 i)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $3$ by $3$ .

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

Step 3.1.2

Multiply $- 2$ by $3$ .

$\sqrt{9 - 6 i + 2 i \cdot 3 + 2 i (- 2 i)}$

Step 3.1.3

Multiply $3$ by $2$ .

$\sqrt{9 - 6 i + 6 i + 2 i (- 2 i)}$

Step 3.1.4

Multiply $2 i (- 2 i)$ .

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Step 3.1.4.1

Multiply $- 2$ by $2$ .

$\sqrt{9 - 6 i + 6 i - 4 i i}$

Step 3.1.4.2

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

$\sqrt{9 - 6 i + 6 i - 4 (i^{1} i)}$

Step 3.1.4.3

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

$\sqrt{9 - 6 i + 6 i - 4 (i^{1} i^{1})}$

Step 3.1.4.4

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

$\sqrt{9 - 6 i + 6 i - 4 i^{1 + 1}}$

Step 3.1.4.5

Add $1$ and $1$ .

$\sqrt{9 - 6 i + 6 i - 4 i^{2}}$

Step 3.1.5

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

$\sqrt{9 - 6 i + 6 i - 4 \cdot - 1}$

Step 3.1.6

Multiply $- 4$ by $- 1$ .

$\sqrt{9 - 6 i + 6 i + 4}$

Step 3.2

Add $9$ and $4$ .

$\sqrt{13 - 6 i + 6 i}$

Step 3.3

Add $- 6 i$ and $6 i$ .

$\sqrt{13 + 0}$

Step 3.4

Add $13$ and $0$ .

$\sqrt{13}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{13}$

Decimal Form:

$3.60555127 \dots$