Finite Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(\sqrt{2} + \sqrt{5}) (\sqrt{2} + \sqrt{5})}{- 3}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{2} + \sqrt{5}$ to the power of $1$ .

$\frac{{(\sqrt{2} + \sqrt{5})}^{1} (\sqrt{2} + \sqrt{5})}{- 3}$

Step 3.2

Raise $\sqrt{2} + \sqrt{5}$ to the power of $1$ .

$\frac{{(\sqrt{2} + \sqrt{5})}^{1} {(\sqrt{2} + \sqrt{5})}^{1}}{- 3}$

Step 3.3

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

$\frac{{(\sqrt{2} + \sqrt{5})}^{1 + 1}}{- 3}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{2} + \sqrt{5})}^{2}}{- 3}$

Step 4

Rewrite ${(\sqrt{2} + \sqrt{5})}^{2}$ as $(\sqrt{2} + \sqrt{5}) (\sqrt{2} + \sqrt{5})$ .

$\frac{(\sqrt{2} + \sqrt{5}) (\sqrt{2} + \sqrt{5})}{- 3}$

Step 5

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

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{2} (\sqrt{2} + \sqrt{5}) + \sqrt{5} (\sqrt{2} + \sqrt{5})}{- 3}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} \sqrt{5} + \sqrt{5} (\sqrt{2} + \sqrt{5})}{- 3}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{2} + \sqrt{2} \sqrt{5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

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

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 2} + \sqrt{2} \sqrt{5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.2

Multiply $2$ by $2$ .

$\frac{\sqrt{4} + \sqrt{2} \sqrt{5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.3

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

$\frac{\sqrt{2^{2}} + \sqrt{2} \sqrt{5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.4

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

$\frac{2 + \sqrt{2} \sqrt{5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.5

Combine using the product rule for radicals.

$\frac{2 + \sqrt{2 \cdot 5} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.6

Multiply $2$ by $5$ .

$\frac{2 + \sqrt{10} + \sqrt{5} \sqrt{2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.7

Combine using the product rule for radicals.

$\frac{2 + \sqrt{10} + \sqrt{5 \cdot 2} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.8

Multiply $5$ by $2$ .

$\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5} \sqrt{5}}{- 3}$

Step 6.1.9

Combine using the product rule for radicals.

$\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5 \cdot 5}}{- 3}$

Step 6.1.10

Multiply $5$ by $5$ .

$\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{25}}{- 3}$

Step 6.1.11

Rewrite $25$ as $5^{2}$ .

$\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5^{2}}}{- 3}$

Step 6.1.12

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

$\frac{2 + \sqrt{10} + \sqrt{10} + 5}{- 3}$

Step 6.2

Add $2$ and $5$ .

$\frac{7 + \sqrt{10} + \sqrt{10}}{- 3}$

Step 6.3

Add $\sqrt{10}$ and $\sqrt{10}$ .

$\frac{7 + 2 \sqrt{10}}{- 3}$

Step 7

Move the negative in front of the fraction.

$- \frac{7 + 2 \sqrt{10}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{7 + 2 \sqrt{10}}{3}$

Decimal Form:

$- 4.44151844 \dots$