Algebra Examples

$\frac{8 - 8 i \sqrt{5}}{6 + 2 i \sqrt{5}}$

Step 1

Cancel the common factor of $8 - 8 i \sqrt{5}$ and $6 + 2 i \sqrt{5}$ .

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Step 1.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4 - 8 i \sqrt{5}}{6 + 2 i \sqrt{5}}$

Step 1.2

Factor $2$ out of $- 8 i \sqrt{5}$ .

$\frac{2 \cdot 4 + 2 (- 4 i \sqrt{5})}{6 + 2 i \sqrt{5}}$

Step 1.3

Factor $2$ out of $2 (4) + 2 (- 4 i \sqrt{5})$ .

$\frac{2 (4 - 4 i \sqrt{5})}{6 + 2 i \sqrt{5}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $6$ .

$\frac{2 (4 - 4 i \sqrt{5})}{2 (3) + 2 i \sqrt{5}}$

Step 1.4.2

Factor $2$ out of $2 i \sqrt{5}$ .

$\frac{2 (4 - 4 i \sqrt{5})}{2 (3) + 2 (i \sqrt{5})}$

Step 1.4.3

Factor $2$ out of $2 (3) + 2 (i \sqrt{5})$ .

$\frac{2 (4 - 4 i \sqrt{5})}{2 (3 + i \sqrt{5})}$

Step 1.4.4

Cancel the common factor.

$\frac{2 (4 - 4 i \sqrt{5})}{2 (3 + i \sqrt{5})}$

Step 1.4.5

Rewrite the expression.

$\frac{4 - 4 i \sqrt{5}}{3 + i \sqrt{5}}$

Step 2

Multiply $\frac{4 - 4 i \sqrt{5}}{3 + i \sqrt{5}}$ by $\frac{3 - i \sqrt{5}}{3 - i \sqrt{5}}$ .

$\frac{4 - 4 i \sqrt{5}}{3 + i \sqrt{5}} \cdot \frac{3 - i \sqrt{5}}{3 - i \sqrt{5}}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{4 - 4 i \sqrt{5}}{3 + i \sqrt{5}}$ by $\frac{3 - i \sqrt{5}}{3 - i \sqrt{5}}$ .

$\frac{(4 - 4 i \sqrt{5}) (3 - i \sqrt{5})}{(3 + i \sqrt{5}) (3 - i \sqrt{5})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(4 - 4 i \sqrt{5}) (3 - i \sqrt{5})}{9 - 3 (i \sqrt{5}) + i \sqrt{5} \cdot 3 - i^{2} {\sqrt{5}}^{2}}$

Step 3.3

Simplify.

$\frac{(4 - 4 i \sqrt{5}) (3 - i \sqrt{5})}{14}$

Step 3.4

Cancel the common factor of $4 - 4 i \sqrt{5}$ and $14$ .

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Step 3.4.1

Factor $2$ out of $(4 - 4 i \sqrt{5}) (3 - i \sqrt{5})$ .

$\frac{2 ((2 - 2 i \sqrt{5}) (3 - i \sqrt{5}))}{14}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $14$ .

$\frac{2 ((2 - 2 i \sqrt{5}) (3 - i \sqrt{5}))}{2 (7)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 ((2 - 2 i \sqrt{5}) (3 - i \sqrt{5}))}{2 \cdot 7}$

Step 3.4.2.3

Rewrite the expression.

$\frac{(2 - 2 i \sqrt{5}) (3 - i \sqrt{5})}{7}$

Step 4

Expand $(2 - 2 i \sqrt{5}) (3 - i \sqrt{5})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$\frac{2 (3 - i \sqrt{5}) - 2 i \sqrt{5} (3 - i \sqrt{5})}{7}$

Step 4.2

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- i \sqrt{5}) - 2 i \sqrt{5} (3 - i \sqrt{5})}{7}$

Step 4.3

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- i \sqrt{5}) - 2 i \sqrt{5} \cdot 3 - 2 i \sqrt{5} (- i \sqrt{5})}{7}$

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 $2$ by $3$ .

$\frac{6 + 2 (- i \sqrt{5}) - 2 i \sqrt{5} \cdot 3 - 2 i \sqrt{5} (- i \sqrt{5})}{7}$

Step 5.1.2

Multiply $- 1$ by $2$ .

$\frac{6 - 2 (i \sqrt{5}) - 2 i \sqrt{5} \cdot 3 - 2 i \sqrt{5} (- i \sqrt{5})}{7}$

Step 5.1.3

Multiply $3$ by $- 2$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 i \sqrt{5} (- i \sqrt{5})}{7}$

Step 5.1.4

Multiply $- 2 i \sqrt{5} (- i \sqrt{5})$ .

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Step 5.1.4.1

Multiply $- 1$ by $- 2$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i \sqrt{5} (i \sqrt{5})}{7}$

Step 5.1.4.2

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 (i^{1} i) \sqrt{5} \sqrt{5}}{7}$

Step 5.1.4.3

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 (i^{1} i^{1}) \sqrt{5} \sqrt{5}}{7}$

Step 5.1.4.4

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{1 + 1} \sqrt{5} \sqrt{5}}{7}$

Step 5.1.4.5

Add $1$ and $1$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{2} \sqrt{5} \sqrt{5}}{7}$

Step 5.1.4.6

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{2} ({\sqrt{5}}^{1} \sqrt{5})}{7}$

Step 5.1.4.7

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{7}$

Step 5.1.4.8

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{2} {\sqrt{5}}^{1 + 1}}{7}$

Step 5.1.4.9

Add $1$ and $1$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 i^{2} {\sqrt{5}}^{2}}{7}$

Step 5.1.5

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} + 2 \cdot - 1 {\sqrt{5}}^{2}}{7}$

Step 5.1.6

Multiply $2$ by $- 1$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 {\sqrt{5}}^{2}}{7}$

Step 5.1.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 {(5^{\frac{1}{2}})}^{2}}{7}$

Step 5.1.7.2

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 \cdot 5^{\frac{1}{2} \cdot 2}}{7}$

Step 5.1.7.3

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

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 \cdot 5^{\frac{2}{2}}}{7}$

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.

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 \cdot 5^{\frac{2}{2}}}{7}$

Step 5.1.7.4.2

Rewrite the expression.

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 \cdot 5^{1}}{7}$

Step 5.1.7.5

Evaluate the exponent.

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 2 \cdot 5}{7}$

Step 5.1.8

Multiply $- 2$ by $5$ .

$\frac{6 - 2 i \sqrt{5} - 6 i \sqrt{5} - 10}{7}$

Step 5.2

Subtract $10$ from $6$ .

$\frac{- 2 i \sqrt{5} - 6 i \sqrt{5} - 4}{7}$

Step 5.3

Subtract $6 i \sqrt{5}$ from $- 2 i \sqrt{5}$ .

$\frac{- 8 i \sqrt{5} - 4}{7}$

Step 6

Reorder $- 8 i \sqrt{5}$ and $- 4$ .

$\frac{- 4 - 8 i \sqrt{5}}{7}$

Step 7

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) - 8 i \sqrt{5}}{7}$

Step 8

Factor $- 1$ out of $- 8 i \sqrt{5}$ .

$\frac{- 1 (4) - (8 i \sqrt{5})}{7}$

Step 9

Factor $- 1$ out of $- 1 (4) - (8 i \sqrt{5})$ .

$\frac{- 1 (4 + 8 i \sqrt{5})}{7}$

Step 10

Move the negative in front of the fraction.

$- \frac{4 + 8 i \sqrt{5}}{7}$