Algebra Examples

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

Step 1

Multiply $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$ by $\frac{\sqrt{2} - 1}{\sqrt{2} - 1}$ .

$\frac{\sqrt{2} - 1}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$ by $\frac{\sqrt{2} - 1}{\sqrt{2} - 1}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(\sqrt{2} - 1) (\sqrt{2} - 1)}{1}$

Step 2.4

Divide $(\sqrt{2} - 1) (\sqrt{2} - 1)$ by $1$ .

$(\sqrt{2} - 1) (\sqrt{2} - 1)$

Step 3

Expand $(\sqrt{2} - 1) (\sqrt{2} - 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{2} (\sqrt{2} - 1) - 1 (\sqrt{2} - 1)$

Step 3.2

Apply the distributive property.

$\sqrt{2} \sqrt{2} + \sqrt{2} \cdot - 1 - 1 (\sqrt{2} - 1)$

Step 3.3

Apply the distributive property.

$\sqrt{2} \sqrt{2} + \sqrt{2} \cdot - 1 - 1 \sqrt{2} - 1 \cdot - 1$

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

Combine using the product rule for radicals.

$\sqrt{2 \cdot 2} + \sqrt{2} \cdot - 1 - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.2

Multiply $2$ by $2$ .

$\sqrt{4} + \sqrt{2} \cdot - 1 - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.3

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

$\sqrt{2^{2}} + \sqrt{2} \cdot - 1 - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.4

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

$2 + \sqrt{2} \cdot - 1 - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.5

Move $- 1$ to the left of $\sqrt{2}$ .

$2 - 1 \cdot \sqrt{2} - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.6

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$2 - \sqrt{2} - 1 \sqrt{2} - 1 \cdot - 1$

Step 4.1.7

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$2 - \sqrt{2} - \sqrt{2} - 1 \cdot - 1$

Step 4.1.8

Multiply $- 1$ by $- 1$ .

$2 - \sqrt{2} - \sqrt{2} + 1$

Step 4.2

Add $2$ and $1$ .

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

Step 4.3

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

$3 - 2 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$3 - 2 \sqrt{2}$

Decimal Form:

$0.17157287 \dots$