Basic Math Examples

$(\sqrt{13} - \sqrt{12 i}) (\sqrt{3} + \sqrt{12 i})$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $12 i$ as $2^{2} \cdot (3 i)$ .

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Step 1.1.1.1

Factor $4$ out of $12$ .

$(\sqrt{13} - \sqrt{4 (3) i}) (\sqrt{3} + \sqrt{12 i})$

Step 1.1.1.2

Rewrite $4$ as $2^{2}$ .

$(\sqrt{13} - \sqrt{2^{2} \cdot 3 i}) (\sqrt{3} + \sqrt{12 i})$

Step 1.1.1.3

Add parentheses.

$(\sqrt{13} - \sqrt{2^{2} \cdot (3 i)}) (\sqrt{3} + \sqrt{12 i})$

Step 1.1.2

Pull terms out from under the radical.

$(\sqrt{13} - (2 \sqrt{3 i})) (\sqrt{3} + \sqrt{12 i})$

Step 1.1.3

Multiply $2$ by $- 1$ .

$(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + \sqrt{12 i})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $12 i$ as $2^{2} \cdot (3 i)$ .

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Step 1.2.1.1

Factor $4$ out of $12$ .

$(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + \sqrt{4 (3) i})$

Step 1.2.1.2

Rewrite $4$ as $2^{2}$ .

$(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + \sqrt{2^{2} \cdot 3 i})$

Step 1.2.1.3

Add parentheses.

$(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + \sqrt{2^{2} \cdot (3 i)})$

Step 1.2.2

Pull terms out from under the radical.

$(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + 2 \sqrt{3 i})$

Step 2

Expand $(\sqrt{13} - 2 \sqrt{3 i}) (\sqrt{3} + 2 \sqrt{3 i})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{13} (\sqrt{3} + 2 \sqrt{3 i}) - 2 \sqrt{3 i} (\sqrt{3} + 2 \sqrt{3 i})$

Step 2.2

Apply the distributive property.

$\sqrt{13} \sqrt{3} + \sqrt{13} (2 \sqrt{3 i}) - 2 \sqrt{3 i} (\sqrt{3} + 2 \sqrt{3 i})$

Step 2.3

Apply the distributive property.

$\sqrt{13} \sqrt{3} + \sqrt{13} (2 \sqrt{3 i}) - 2 \sqrt{3 i} \sqrt{3} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3

Simplify each term.

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Step 3.1

Combine using the product rule for radicals.

$\sqrt{13 \cdot 3} + \sqrt{13} (2 \sqrt{3 i}) - 2 \sqrt{3 i} \sqrt{3} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.2

Multiply $13$ by $3$ .

$\sqrt{39} + \sqrt{13} (2 \sqrt{3 i}) - 2 \sqrt{3 i} \sqrt{3} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.3

Multiply $\sqrt{13} (2 \sqrt{3 i})$ .

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Step 3.3.1

Combine using the product rule for radicals.

$\sqrt{39} + 2 \sqrt{13 (3 i)} - 2 \sqrt{3 i} \sqrt{3} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.3.2

Multiply $13$ by $3$ .

$\sqrt{39} + 2 \sqrt{39 i} - 2 \sqrt{3 i} \sqrt{3} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.4

Multiply $- 2 \sqrt{3 i} \sqrt{3}$ .

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Step 3.4.1

Combine using the product rule for radicals.

$\sqrt{39} + 2 \sqrt{39 i} - 2 \sqrt{3 (3 i)} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.4.2

Multiply $3$ by $3$ .

$\sqrt{39} + 2 \sqrt{39 i} - 2 \sqrt{9 i} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.5

Rewrite $9$ as $3^{2}$ .

$\sqrt{39} + 2 \sqrt{39 i} - 2 \sqrt{3^{2} i} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.6

Pull terms out from under the radical.

$\sqrt{39} + 2 \sqrt{39 i} - 2 (3 \sqrt{i}) - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.7

Multiply $3$ by $- 2$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 2 \sqrt{3 i} (2 \sqrt{3 i})$

Step 3.8

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

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Step 3.8.1

Multiply $2$ by $- 2$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 \sqrt{3 i} \sqrt{3 i}$

Step 3.8.2

Raise $\sqrt{3 i}$ to the power of $1$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 ({\sqrt{3 i}}^{1} \sqrt{3 i})$

Step 3.8.3

Raise $\sqrt{3 i}$ to the power of $1$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 ({\sqrt{3 i}}^{1} {\sqrt{3 i}}^{1})$

Step 3.8.4

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

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {\sqrt{3 i}}^{1 + 1}$

Step 3.8.5

Add $1$ and $1$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {\sqrt{3 i}}^{2}$

Step 3.9

Rewrite ${\sqrt{3 i}}^{2}$ as $3 i$ .

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Step 3.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 i}$ as ${(3 i)}^{\frac{1}{2}}$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {({(3 i)}^{\frac{1}{2}})}^{2}$

Step 3.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {(3 i)}^{\frac{1}{2} \cdot 2}$

Step 3.9.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {(3 i)}^{\frac{2}{2}}$

Step 3.9.4

Cancel the common factor of $2$ .

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Step 3.9.4.1

Cancel the common factor.

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {(3 i)}^{\frac{2}{2}}$

Step 3.9.4.2

Rewrite the expression.

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 {(3 i)}^{1}$

Step 3.9.5

Simplify.

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 4 (3 i)$

Step 3.10

Multiply $3$ by $- 4$ .

$\sqrt{39} + 2 \sqrt{39 i} - 6 \sqrt{i} - 12 i$