Linear Algebra Examples

\frac{8}{\sqrt{- 7} - \sqrt{3}}

$\frac{8}{\sqrt{- 7} - \sqrt{3}}$

Step 1

Pull out imaginary unit

i

$i$ .

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Step 1.1

Rewrite

- 7

$- 7$ as

- 1 (7)

$- 1 (7)$ .

\frac{8}{\sqrt{- 1 (7)} - \sqrt{3}}

$\frac{8}{\sqrt{- 1 (7)} - \sqrt{3}}$

Step 1.2

Rewrite

\sqrt{- 1 (7)}

$\sqrt{- 1 (7)}$ as

\sqrt{- 1} \cdot \sqrt{7}

$\sqrt{- 1} \cdot \sqrt{7}$ .

\frac{8}{\sqrt{- 1} \cdot \sqrt{7} - \sqrt{3}}

$\frac{8}{\sqrt{- 1} \cdot \sqrt{7} - \sqrt{3}}$

Step 1.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}

$\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}$

\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}

$\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}$

Step 2

Multiply

\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}

$\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}$ by

\frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}

$\frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}$ .

\frac{8}{i \cdot \sqrt{7} - \sqrt{3}} \cdot \frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}

$\frac{8}{i \cdot \sqrt{7} - \sqrt{3}} \cdot \frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}$

Step 3

Multiply

\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}

$\frac{8}{i \cdot \sqrt{7} - \sqrt{3}}$ by

\frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}

$\frac{i \cdot \sqrt{7} + \sqrt{3}}{i \cdot \sqrt{7} + \sqrt{3}}$ .

\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{(i \cdot \sqrt{7} - \sqrt{3}) (i \cdot \sqrt{7} + \sqrt{3})}

$\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{(i \cdot \sqrt{7} - \sqrt{3}) (i \cdot \sqrt{7} + \sqrt{3})}$

Step 4

Expand the denominator using the FOIL method.

\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{i^{2} {\sqrt{7}}^{2} + i \sqrt{21} - \sqrt{21} i - {\sqrt{3}}^{2}}

$\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{i^{2} {\sqrt{7}}^{2} + i \sqrt{21} - \sqrt{21} i - {\sqrt{3}}^{2}}$

Step 5

Simplify.

\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{- 10}

$\frac{8 (i \cdot \sqrt{7} + \sqrt{3})}{- 10}$

Step 6

Cancel the common factor of

8

$8$ and

- 10

$- 10$ .

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Step 6.1

Factor

2

$2$ out of

8 (i \cdot \sqrt{7} + \sqrt{3})

$8 (i \cdot \sqrt{7} + \sqrt{3})$ .

\frac{2 (4 (i \cdot \sqrt{7} + \sqrt{3}))}{- 10}

$\frac{2 (4 (i \cdot \sqrt{7} + \sqrt{3}))}{- 10}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor

2

$2$ out of

- 10

$- 10$ .

\frac{2 (4 (i \cdot \sqrt{7} + \sqrt{3}))}{2 (- 5)}

$\frac{2 (4 (i \cdot \sqrt{7} + \sqrt{3}))}{2 (- 5)}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (4 (i \cdot \sqrt{7} + \sqrt{3}))}{2 \cdot - 5}$

Step 6.2.3

Rewrite the expression.

$\frac{4 (i \cdot \sqrt{7} + \sqrt{3})}{- 5}$

Step 7

Simplify the expression.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{4 (i \cdot \sqrt{7} + \sqrt{3})}{5}$

Step 7.2

Reorder

$i \sqrt{7}$ and

$\sqrt{3}$ .

$- \frac{4 (\sqrt{3} + i \sqrt{7})}{5}$