Basic Math Examples

$\frac{- 2 + \sqrt{2}}{5 + \sqrt{2}}$

Step 1

Multiply $\frac{- 2 + \sqrt{2}}{5 + \sqrt{2}}$ by $\frac{5 - \sqrt{2}}{5 - \sqrt{2}}$ .

$\frac{- 2 + \sqrt{2}}{5 + \sqrt{2}} \cdot \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{- 2 + \sqrt{2}}{5 + \sqrt{2}}$ by $\frac{5 - \sqrt{2}}{5 - \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(- 2 + \sqrt{2}) (5 - \sqrt{2})}{25 - 5 \sqrt{2} + \sqrt{2} \cdot 5 - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(- 2 + \sqrt{2}) (5 - \sqrt{2})}{23}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{- 2 (5 - \sqrt{2}) + \sqrt{2} (5 - \sqrt{2})}{23}$

Step 3.2

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 (- \sqrt{2}) + \sqrt{2} (5 - \sqrt{2})}{23}$

Step 3.3

Apply the distributive property.

$\frac{- 2 \cdot 5 - 2 (- \sqrt{2}) + \sqrt{2} \cdot 5 + \sqrt{2} (- \sqrt{2})}{23}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- 2$ by $5$ .

$\frac{- 10 - 2 (- \sqrt{2}) + \sqrt{2} \cdot 5 + \sqrt{2} (- \sqrt{2})}{23}$

Step 4.1.2

Multiply $- 1$ by $- 2$ .

$\frac{- 10 + 2 \sqrt{2} + \sqrt{2} \cdot 5 + \sqrt{2} (- \sqrt{2})}{23}$

Step 4.1.3

Move $5$ to the left of $\sqrt{2}$ .

$\frac{- 10 + 2 \sqrt{2} + 5 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{2})}{23}$

Step 4.1.4

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

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Step 4.1.4.1

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}{23}$

Step 4.1.4.2

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{23}$

Step 4.1.4.3

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - {\sqrt{2}}^{1 + 1}}{23}$

Step 4.1.4.4

Add $1$ and $1$ .

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - {\sqrt{2}}^{2}}{23}$

Step 4.1.5

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

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Step 4.1.5.1

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}{23}$

Step 4.1.5.2

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}{23}$

Step 4.1.5.3

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

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 2^{\frac{2}{2}}}{23}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 2^{\frac{2}{2}}}{23}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 2^{1}}{23}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 1 \cdot 2}{23}$

Step 4.1.6

Multiply $- 1$ by $2$ .

$\frac{- 10 + 2 \sqrt{2} + 5 \sqrt{2} - 2}{23}$

Step 4.2

Subtract $2$ from $- 10$ .

$\frac{- 12 + 2 \sqrt{2} + 5 \sqrt{2}}{23}$

Step 4.3

Add $2 \sqrt{2}$ and $5 \sqrt{2}$ .

$\frac{- 12 + 7 \sqrt{2}}{23}$

Step 5

Rewrite $- 12$ as $- 1 (12)$ .

$\frac{- 1 (12) + 7 \sqrt{2}}{23}$

Step 6

Factor $- 1$ out of $7 \sqrt{2}$ .

$\frac{- 1 (12) - (- 7 \sqrt{2})}{23}$

Step 7

Factor $- 1$ out of $- 1 (12) - (- 7 \sqrt{2})$ .

$\frac{- 1 (12 - 7 \sqrt{2})}{23}$

Step 8

Move the negative in front of the fraction.

$- \frac{12 - 7 \sqrt{2}}{23}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{12 - 7 \sqrt{2}}{23}$

Decimal Form:

$- 0.09132630 \dots$