Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{4 (2)}}$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + \sqrt{2}) (\sqrt{5} + 2 \sqrt{2})}{{\sqrt{5}}^{2} + 2 \sqrt{10} - 2 \sqrt{10} - 4 {\sqrt{2}}^{2}}$

Step 3.3

Simplify.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Combine using the product rule for radicals.

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

Step 5.1.2

Multiply $5$ by $5$ .

$\frac{\sqrt{25} + \sqrt{5} (2 \sqrt{2}) + \sqrt{2} \sqrt{5} + \sqrt{2} (2 \sqrt{2})}{- 3}$

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Multiply $\sqrt{5} (2 \sqrt{2})$ .

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Step 5.1.5.1

Combine using the product rule for radicals.

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

Step 5.1.5.2

Multiply $5$ by $2$ .

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

Step 5.1.6

Combine using the product rule for radicals.

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

Step 5.1.7

Multiply $2$ by $5$ .

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

Step 5.1.8

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

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Step 5.1.8.1

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

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 ({\sqrt{2}}^{1} \sqrt{2})}{- 3}$

Step 5.1.8.2

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

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{- 3}$

Step 5.1.8.3

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

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 {\sqrt{2}}^{1 + 1}}{- 3}$

Step 5.1.8.4

Add $1$ and $1$ .

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

Step 5.1.9

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

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Step 5.1.9.1

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

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 {(2^{\frac{1}{2}})}^{2}}{- 3}$

Step 5.1.9.2

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

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 \cdot 2^{\frac{1}{2} \cdot 2}}{- 3}$

Step 5.1.9.3

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

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

Step 5.1.9.4

Cancel the common factor of $2$ .

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Step 5.1.9.4.1

Cancel the common factor.

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

Step 5.1.9.4.2

Rewrite the expression.

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 2 \cdot 2^{1}}{- 3}$

Step 5.1.9.5

Evaluate the exponent.

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

Step 5.1.10

Multiply $2$ by $2$ .

$\frac{5 + 2 \sqrt{10} + \sqrt{10} + 4}{- 3}$

Step 5.2

Add $5$ and $4$ .

$\frac{9 + 2 \sqrt{10} + \sqrt{10}}{- 3}$

Step 5.3

Add $2 \sqrt{10}$ and $\sqrt{10}$ .

$\frac{9 + 3 \sqrt{10}}{- 3}$

Step 6

Cancel the common factor of $9 + 3 \sqrt{10}$ and $- 3$ .

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Step 6.1

Factor $3$ out of $9$ .

$\frac{3 (3) + 3 \sqrt{10}}{- 3}$

Step 6.2

Factor $3$ out of $3 \sqrt{10}$ .

$\frac{3 (3) + 3 (\sqrt{10})}{- 3}$

Step 6.3

Factor $3$ out of $3 (3) + 3 (\sqrt{10})$ .

$\frac{3 (3 + \sqrt{10})}{- 3}$

Step 6.4

Move the negative one from the denominator of $\frac{3 + \sqrt{10}}{- 1}$ .

$- 1 \cdot (3 + \sqrt{10})$

Step 7

Rewrite $- 1 \cdot (3 + \sqrt{10})$ as $- (3 + \sqrt{10})$ .

$- (3 + \sqrt{10})$

Step 8

Apply the distributive property.

$- 1 \cdot 3 - \sqrt{10}$

Step 9

Multiply $- 1$ by $3$ .

$- 3 - \sqrt{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 3 - \sqrt{10}$

Decimal Form:

$- 6.16227766 \dots$