Precalculus Examples

$\frac{1 + 2 \sqrt{3 i}}{1 + \sqrt{3 i}}$

Step 1

Multiply $\frac{1 + 2 \sqrt{3 i}}{1 + \sqrt{3 i}}$ by $\frac{1 - \sqrt{3 i}}{1 - \sqrt{3 i}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 + 2 \sqrt{3 i}}{1 + \sqrt{3 i}}$ by $\frac{1 - \sqrt{3 i}}{1 - \sqrt{3 i}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(1 + 2 \sqrt{3 i}) (1 - \sqrt{3 i})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $1$ by $1$ .

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

Step 4.1.2

Multiply $- \sqrt{3 i}$ by $1$ .

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

Step 4.1.3

Multiply $2$ by $1$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $2$ .

$\frac{1 - \sqrt{3 i} + 2 \sqrt{3 i} - 2 \sqrt{3 i} \sqrt{3 i}}{1 - 3 i}$

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

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

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

Step 4.1.4.5

Add $1$ and $1$ .

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

Step 4.1.5

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

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

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

Step 4.1.5.4.2

Rewrite the expression.

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

Step 4.1.5.5

Simplify.

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

Step 4.1.6

Multiply $3$ by $- 2$ .

$\frac{1 - \sqrt{3 i} + 2 \sqrt{3 i} - 6 i}{1 - 3 i}$

Step 4.2

Add $- \sqrt{3 i}$ and $2 \sqrt{3 i}$ .

$\frac{1 + \sqrt{3 i} - 6 i}{1 - 3 i}$