Algebra Examples

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

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 1.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $1$ by $1$ .

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

Step 5.1.2

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

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

Step 5.1.3

Multiply $2$ by $1$ .

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

Step 5.1.4

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

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Step 5.1.4.1

Multiply $- 1$ by $2$ .

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

Step 5.1.4.2

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

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

Step 5.1.4.3

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

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

Step 5.1.4.4

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

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

Step 5.1.4.5

Add $1$ and $1$ .

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

Step 5.1.4.6

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

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

Step 5.1.4.7

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

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

Step 5.1.4.8

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

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

Step 5.1.4.9

Add $1$ and $1$ .

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

Step 5.1.5

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

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} - 2 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 5.1.6

Multiply $- 2$ by $- 1$ .

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

Step 5.1.7

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

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Step 5.1.7.1

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

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

Step 5.1.7.2

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

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 5.1.7.3

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

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{4}$

Step 5.1.7.4

Cancel the common factor of $2$ .

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Step 5.1.7.4.1

Cancel the common factor.

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{4}$

Step 5.1.7.4.2

Rewrite the expression.

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} + 2 \cdot 3^{1}}{4}$

Step 5.1.7.5

Evaluate the exponent.

$\frac{1 - i \sqrt{3} + 2 i \sqrt{3} + 2 \cdot 3}{4}$

Step 5.1.8

Multiply $2$ by $3$ .

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

Step 5.2

Add $1$ and $6$ .

$\frac{- i \sqrt{3} + 2 i \sqrt{3} + 7}{4}$

Step 5.3

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

$\frac{1 i \sqrt{3} + 7}{4}$

Step 6

Multiply $i$ by $1$ .

$\frac{i \sqrt{3} + 7}{4}$

Step 7

Reorder $i \sqrt{3}$ and $7$ .

$\frac{7 + i \sqrt{3}}{4}$