Precalculus Examples

$(\sqrt{3} + \sqrt{15 i}) (\sqrt{3} - \sqrt{15 i})$

Step 1

Expand $(\sqrt{3} + \sqrt{15 i}) (\sqrt{3} - \sqrt{15 i})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{3} (\sqrt{3} - \sqrt{15 i}) + \sqrt{15 i} (\sqrt{3} - \sqrt{15 i})$

Step 1.2

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} (\sqrt{3} - \sqrt{15 i})$

Step 1.3

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.2

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.3

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

$\sqrt{3^{2}} + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$3 + \sqrt{3} (- \sqrt{15 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.5

Multiply $\sqrt{3} (- \sqrt{15 i})$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

$3 - \sqrt{3 (15 i)} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.5.2

Multiply $3$ by $15$ .

$3 - \sqrt{45 i} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.6

Rewrite $45 i$ as $3^{2} \cdot (5 i)$ .

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Step 2.1.6.1

Factor $9$ out of $45$ .

$3 - \sqrt{9 (5) i} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.6.2

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

$3 - \sqrt{3^{2} \cdot 5 i} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.6.3

Add parentheses.

$3 - \sqrt{3^{2} \cdot (5 i)} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.7

Pull terms out from under the radical.

$3 - (3 \sqrt{5 i}) + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.8

Multiply $3$ by $- 1$ .

$3 - 3 \sqrt{5 i} + \sqrt{15 i} \sqrt{3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.9

Combine using the product rule for radicals.

$3 - 3 \sqrt{5 i} + \sqrt{15 i \cdot 3} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.10

Multiply $3$ by $15$ .

$3 - 3 \sqrt{5 i} + \sqrt{45 i} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.11

Rewrite $45 i$ as $3^{2} \cdot (5 i)$ .

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Step 2.1.11.1

Factor $9$ out of $45$ .

$3 - 3 \sqrt{5 i} + \sqrt{9 (5) i} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.11.2

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

$3 - 3 \sqrt{5 i} + \sqrt{3^{2} \cdot 5 i} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.11.3

Add parentheses.

$3 - 3 \sqrt{5 i} + \sqrt{3^{2} \cdot (5 i)} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.12

Pull terms out from under the radical.

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} + \sqrt{15 i} (- \sqrt{15 i})$

Step 2.1.13

Multiply $\sqrt{15 i} (- \sqrt{15 i})$ .

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Step 2.1.13.1

Raise $\sqrt{15 i}$ to the power of $1$ .

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - ({\sqrt{15 i}}^{1} \sqrt{15 i})$

Step 2.1.13.2

Raise $\sqrt{15 i}$ to the power of $1$ .

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - ({\sqrt{15 i}}^{1} {\sqrt{15 i}}^{1})$

Step 2.1.13.3

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

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {\sqrt{15 i}}^{1 + 1}$

Step 2.1.13.4

Add $1$ and $1$ .

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {\sqrt{15 i}}^{2}$

Step 2.1.14

Rewrite ${\sqrt{15 i}}^{2}$ as $15 i$ .

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Step 2.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 i}$ as ${(15 i)}^{\frac{1}{2}}$ .

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {({(15 i)}^{\frac{1}{2}})}^{2}$

Step 2.1.14.2

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

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {(15 i)}^{\frac{1}{2} \cdot 2}$

Step 2.1.14.3

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

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {(15 i)}^{\frac{2}{2}}$

Step 2.1.14.4

Cancel the common factor of $2$ .

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Step 2.1.14.4.1

Cancel the common factor.

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {(15 i)}^{\frac{2}{2}}$

Step 2.1.14.4.2

Rewrite the expression.

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - {(15 i)}^{1}$

Step 2.1.14.5

Simplify.

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - (15 i)$

Step 2.1.15

Multiply $15$ by $- 1$ .

$3 - 3 \sqrt{5 i} + 3 \sqrt{5 i} - 15 i$

Step 2.2

Add $- 3 \sqrt{5 i}$ and $3 \sqrt{5 i}$ .

$3 + 0 - 15 i$

Step 2.3

Add $3$ and $0$ .

$3 - 15 i$