Linear Algebra Examples

$\frac{- 2 + \sqrt{8}}{- 3 - \sqrt{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(- 2 + 2 \sqrt{2}) (- 3 + \sqrt{2})}{9 - 3 \sqrt{2} + 3 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 3.3

Simplify.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

$\frac{- 2 \cdot - 3 - 2 \sqrt{2} + 2 \sqrt{2} \cdot - 3 + 2 \sqrt{2} \sqrt{2}}{7}$

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 $- 3$ .

$\frac{6 - 2 \sqrt{2} + 2 \sqrt{2} \cdot - 3 + 2 \sqrt{2} \sqrt{2}}{7}$

Step 5.1.2

Multiply $- 3$ by $2$ .

$\frac{6 - 2 \sqrt{2} - 6 \sqrt{2} + 2 \sqrt{2} \sqrt{2}}{7}$

Step 5.1.3

Multiply $2 \sqrt{2} \sqrt{2}$ .

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Step 5.1.3.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.1.3.3

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

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

Step 5.1.3.4

Add $1$ and $1$ .

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

Step 5.1.4

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

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Step 5.1.4.1

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

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

Step 5.1.4.2

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

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

Step 5.1.4.3

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

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

Step 5.1.4.4

Cancel the common factor of $2$ .

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Step 5.1.4.4.1

Cancel the common factor.

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

Step 5.1.4.4.2

Rewrite the expression.

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

Step 5.1.4.5

Evaluate the exponent.

$\frac{6 - 2 \sqrt{2} - 6 \sqrt{2} + 2 \cdot 2}{7}$

Step 5.1.5

Multiply $2$ by $2$ .

$\frac{6 - 2 \sqrt{2} - 6 \sqrt{2} + 4}{7}$

Step 5.2

Add $6$ and $4$ .

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

Step 5.3

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

$\frac{10 - 8 \sqrt{2}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{10 - 8 \sqrt{2}}{7}$

Decimal Form:

$- 0.18767264 \dots$