Algebra Examples

$\frac{5 + \sqrt{2}}{8 - \sqrt{2}}$

Step 1

Multiply $\frac{5 + \sqrt{2}}{8 - \sqrt{2}}$ by $\frac{8 + \sqrt{2}}{8 + \sqrt{2}}$ .

$\frac{5 + \sqrt{2}}{8 - \sqrt{2}} \cdot \frac{8 + \sqrt{2}}{8 + \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 + \sqrt{2}}{8 - \sqrt{2}}$ by $\frac{8 + \sqrt{2}}{8 + \sqrt{2}}$ .

$\frac{(5 + \sqrt{2}) (8 + \sqrt{2})}{(8 - \sqrt{2}) (8 + \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 + \sqrt{2}) (8 + \sqrt{2})}{64 + 8 \sqrt{2} - 8 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 + \sqrt{2}) (8 + \sqrt{2})}{62}$

Step 3

Expand $(5 + \sqrt{2}) (8 + \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 (8 + \sqrt{2}) + \sqrt{2} (8 + \sqrt{2})}{62}$

Step 3.2

Apply the distributive property.

$\frac{5 \cdot 8 + 5 \sqrt{2} + \sqrt{2} (8 + \sqrt{2})}{62}$

Step 3.3

Apply the distributive property.

$\frac{5 \cdot 8 + 5 \sqrt{2} + \sqrt{2} \cdot 8 + \sqrt{2} \sqrt{2}}{62}$

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 $5$ by $8$ .

$\frac{40 + 5 \sqrt{2} + \sqrt{2} \cdot 8 + \sqrt{2} \sqrt{2}}{62}$

Step 4.1.2

Move $8$ to the left of $\sqrt{2}$ .

$\frac{40 + 5 \sqrt{2} + 8 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{62}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{40 + 5 \sqrt{2} + 8 \sqrt{2} + \sqrt{2 \cdot 2}}{62}$

Step 4.1.4

Multiply $2$ by $2$ .

$\frac{40 + 5 \sqrt{2} + 8 \sqrt{2} + \sqrt{4}}{62}$

Step 4.1.5

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

$\frac{40 + 5 \sqrt{2} + 8 \sqrt{2} + \sqrt{2^{2}}}{62}$

Step 4.1.6

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

$\frac{40 + 5 \sqrt{2} + 8 \sqrt{2} + 2}{62}$

Step 4.2

Add $40$ and $2$ .

$\frac{42 + 5 \sqrt{2} + 8 \sqrt{2}}{62}$

Step 4.3

Add $5 \sqrt{2}$ and $8 \sqrt{2}$ .

$\frac{42 + 13 \sqrt{2}}{62}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{42 + 13 \sqrt{2}}{62}$

Decimal Form:

$0.97394800 \dots$