Finite Math Examples

$\frac{\sqrt{7} + i \sqrt{7}}{(1 - i \sqrt{3}) (2 - 2 i)}$

Step 1

Multiply $\frac{\sqrt{7} + i \sqrt{7}}{(1 - i \sqrt{3}) (2 - 2 i)}$ by $\frac{1 + i \sqrt{3}}{1 + i \sqrt{3}}$ .

$\frac{\sqrt{7} + i \sqrt{7}}{(1 - i \sqrt{3}) (2 - 2 i)} \cdot \frac{1 + i \sqrt{3}}{1 + i \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{7} + i \sqrt{7}}{(1 - i \sqrt{3}) (2 - 2 i)}$ by $\frac{1 + i \sqrt{3}}{1 + i \sqrt{3}}$ .

$\frac{(\sqrt{7} + i \sqrt{7}) (1 + i \sqrt{3})}{(1 - i \sqrt{3}) (2 - 2 i) (1 + i \sqrt{3})}$

Step 2.2

Reorder.

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Step 2.2.1

Move $1 - i \sqrt{3}$ .

$\frac{(\sqrt{7} + i \sqrt{7}) (1 + i \sqrt{3})}{(2 - 2 i) ((1 - i \sqrt{3}) (1 + i \sqrt{3}))}$

Step 2.2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} + i \sqrt{7}) (1 + i \sqrt{3})}{(2 - 2 i) (1 + \sqrt{3} i - i \sqrt{3} - i^{2} {\sqrt{3}}^{2})}$

Step 2.2.3

Simplify.

$\frac{(\sqrt{7} + i \sqrt{7}) (1 + i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 3

Expand $(\sqrt{7} + i \sqrt{7}) (1 + i \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{7} (1 + i \sqrt{3}) + i \sqrt{7} (1 + i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{7} \cdot 1 + \sqrt{7} (i \sqrt{3}) + i \sqrt{7} (1 + i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{7} \cdot 1 + \sqrt{7} (i \sqrt{3}) + i \sqrt{7} \cdot 1 + i \sqrt{7} (i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $\sqrt{7}$ by $1$ .

$\frac{\sqrt{7} + \sqrt{7} (i \sqrt{3}) + i \sqrt{7} \cdot 1 + i \sqrt{7} (i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 4.1.2

Multiply $\sqrt{7} (i \sqrt{3})$ .

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Step 4.1.2.1

Combine using the product rule for radicals.

$\frac{\sqrt{7} + \sqrt{3 \cdot 7} i + i \sqrt{7} \cdot 1 + i \sqrt{7} (i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 4.1.2.2

Multiply $3$ by $7$ .

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} \cdot 1 + i \sqrt{7} (i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 4.1.3

Multiply $i$ by $1$ .

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i \sqrt{7} (i \sqrt{3})}{(2 - 2 i) \cdot 4}$

Step 4.1.4

Multiply $i \sqrt{7} (i \sqrt{3})$ .

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Step 4.1.4.1

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

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{1} i \sqrt{7} \sqrt{3}}{(2 - 2 i) \cdot 4}$

Step 4.1.4.2

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

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{1} i^{1} \sqrt{7} \sqrt{3}}{(2 - 2 i) \cdot 4}$

Step 4.1.4.3

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

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{1 + 1} \sqrt{7} \sqrt{3}}{(2 - 2 i) \cdot 4}$

Step 4.1.4.4

Add $1$ and $1$ .

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{2} \sqrt{7} \sqrt{3}}{(2 - 2 i) \cdot 4}$

Step 4.1.4.5

Combine using the product rule for radicals.

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{2} \sqrt{3 \cdot 7}}{(2 - 2 i) \cdot 4}$

Step 4.1.4.6

Multiply $3$ by $7$ .

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} + i^{2} \sqrt{21}}{(2 - 2 i) \cdot 4}$

Step 4.1.5

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

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} - 1 \sqrt{21}}{(2 - 2 i) \cdot 4}$

Step 4.1.6

Rewrite $- 1 \sqrt{21}$ as $- \sqrt{21}$ .

$\frac{\sqrt{7} + \sqrt{21} i + i \sqrt{7} - \sqrt{21}}{(2 - 2 i) \cdot 4}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Reorder $\sqrt{7} + \sqrt{21} i + i \sqrt{7}$ and $- \sqrt{21}$ .

$\frac{- \sqrt{21} + \sqrt{7} + \sqrt{21} i + i \sqrt{7}}{(2 - 2 i) \cdot 4}$

Step 4.2.2

Move $4$ to the left of $2 - 2 i$ .

$\frac{- \sqrt{21} + \sqrt{7} + \sqrt{21} i + i \sqrt{7}}{4 \cdot (2 - 2 i)}$

Step 4.2.3

Factor $- 1$ out of $- \sqrt{21}$ .

$\frac{- (\sqrt{21}) + \sqrt{7} + \sqrt{21} i + i \sqrt{7}}{4 (2 - 2 i)}$

Step 4.2.4

Factor $- 1$ out of $\sqrt{7}$ .

$\frac{- (\sqrt{21}) - 1 (- \sqrt{7}) + \sqrt{21} i + i \sqrt{7}}{4 (2 - 2 i)}$

Step 4.2.5

Factor $- 1$ out of $- (\sqrt{21}) - 1 (- \sqrt{7})$ .

$\frac{- (\sqrt{21} - \sqrt{7}) + \sqrt{21} i + i \sqrt{7}}{4 (2 - 2 i)}$

Step 4.2.6

Factor $- 1$ out of $\sqrt{21} i$ .

$\frac{- (\sqrt{21} - \sqrt{7}) - (- \sqrt{21} i) + i \sqrt{7}}{4 (2 - 2 i)}$

Step 4.2.7

Factor $- 1$ out of $- (\sqrt{21} - \sqrt{7}) - (- \sqrt{21} i)$ .

$\frac{- (\sqrt{21} - \sqrt{7} - \sqrt{21} i) + i \sqrt{7}}{4 (2 - 2 i)}$

Step 4.2.8

Factor $- 1$ out of $i \sqrt{7}$ .

$\frac{- (\sqrt{21} - \sqrt{7} - \sqrt{21} i) - (- i \sqrt{7})}{4 (2 - 2 i)}$

Step 4.2.9

Factor $- 1$ out of $- (\sqrt{21} - \sqrt{7} - \sqrt{21} i) - (- i \sqrt{7})$ .

$\frac{- (\sqrt{21} - \sqrt{7} - \sqrt{21} i - i \sqrt{7})}{4 (2 - 2 i)}$

Step 4.2.10

Simplify the expression.

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Step 4.2.10.1

Rewrite $- (\sqrt{21} - \sqrt{7} - \sqrt{21} i - i \sqrt{7})$ as $- 1 (\sqrt{21} - \sqrt{7} - \sqrt{21} i - i \sqrt{7})$ .

$\frac{- 1 (\sqrt{21} - \sqrt{7} - \sqrt{21} i - i \sqrt{7})}{4 (2 - 2 i)}$

Step 4.2.10.2

Move the negative in front of the fraction.

$- \frac{\sqrt{21} - \sqrt{7} - \sqrt{21} i - i \sqrt{7}}{4 (2 - 2 i)}$