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.

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

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 $45$ as $3^{2} \cdot 5$ .

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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 2.1.7

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

Multiply $\sqrt{15} i \sqrt{3}$ .

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Step 2.1.9.1

Combine using the product rule for radicals.

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

Step 2.1.9.2

Multiply $3$ by $15$ .

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

Step 2.1.10

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

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Step 2.1.10.1

Factor $9$ out of $45$ .

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

Step 2.1.10.2

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

Pull terms out from under the radical.

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

Step 2.1.12

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

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Step 2.1.12.1

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

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

Step 2.1.12.2

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

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

Step 2.1.12.3

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

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

Step 2.1.12.4

Add $1$ and $1$ .

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

Step 2.1.12.5

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

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

Step 2.1.12.6

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

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

Step 2.1.12.7

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

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

Step 2.1.12.8

Add $1$ and $1$ .

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

Step 2.1.13

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

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Step 2.1.13.1

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

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

Step 2.1.13.2

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

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

Step 2.1.13.3

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

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

Step 2.1.13.4

Cancel the common factor of $2$ .

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Step 2.1.13.4.1

Cancel the common factor.

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

Step 2.1.13.4.2

Rewrite the expression.

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

Step 2.1.13.5

Evaluate the exponent.

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

Step 2.1.14

Multiply $- 1$ by $15$ .

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

Step 2.1.15

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

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

Step 2.1.16

Multiply $- 15$ by $- 1$ .

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

Step 2.2

Add $3$ and $15$ .

$- 3 \sqrt{5} i + 3 \sqrt{5} i + 18$

Step 2.3

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

$0 + 18$

Step 2.4

Add $0$ and $18$ .

$18$