Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Reorder terms.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

$\frac{{(2 + \sqrt{2})}^{2}}{2}$

Step 4

Rewrite ${(2 + \sqrt{2})}^{2}$ as $(2 + \sqrt{2}) (2 + \sqrt{2})$ .

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

$\frac{2 \cdot 2 + 2 \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}$

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $2$ by $2$ .

$\frac{4 + 2 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}$

Step 6.1.2

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

$\frac{4 + 2 \sqrt{2} + 2 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{2}$

Step 6.1.3

Combine using the product rule for radicals.

$\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{2 \cdot 2}}{2}$

Step 6.1.4

Multiply $2$ by $2$ .

$\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{4}}{2}$

Step 6.1.5

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

$\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + \sqrt{2^{2}}}{2}$

Step 6.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 + 2 \sqrt{2} + 2 \sqrt{2} + 2}{2}$

Step 6.2

Add $4$ and $2$ .

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

Step 6.3

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

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

Step 7

Cancel the common factor of $6 + 4 \sqrt{2}$ and $2$ .

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Step 7.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3 + 4 \sqrt{2}}{2}$

Step 7.2

Factor $2$ out of $4 \sqrt{2}$ .

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

Step 7.3

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

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

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $2$ .

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

Step 7.4.2

Cancel the common factor.

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

Step 7.4.3

Rewrite the expression.

$\frac{3 + 2 \sqrt{2}}{1}$

Step 7.4.4

Divide $3 + 2 \sqrt{2}$ by $1$ .

$3 + 2 \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$3 + 2 \sqrt{2}$

Decimal Form:

$5.82842712 \dots$