Finite Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(\sqrt{5} - 2 \sqrt{2}) (\sqrt{2} + 2 \sqrt{5})}{- 18}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{2} + 2 \sqrt{5}) - 2 \sqrt{2} (\sqrt{2} + 2 \sqrt{5})}{- 18}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{2} + \sqrt{5} (2 \sqrt{5}) - 2 \sqrt{2} (\sqrt{2} + 2 \sqrt{5})}{- 18}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{2} + \sqrt{5} (2 \sqrt{5}) - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

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.

$\frac{\sqrt{5 \cdot 2} + \sqrt{5} (2 \sqrt{5}) - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.2

Multiply $5$ by $2$ .

$\frac{\sqrt{10} + \sqrt{5} (2 \sqrt{5}) - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{\sqrt{10} + 2 ({\sqrt{5}}^{1} \sqrt{5}) - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.3.2

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

$\frac{\sqrt{10} + 2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.3.3

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

$\frac{\sqrt{10} + 2 {\sqrt{5}}^{1 + 1} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.3.4

Add $1$ and $1$ .

$\frac{\sqrt{10} + 2 {\sqrt{5}}^{2} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{10} + 2 {(5^{\frac{1}{2}})}^{2} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4.2

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

$\frac{\sqrt{10} + 2 \cdot 5^{\frac{1}{2} \cdot 2} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4.3

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

$\frac{\sqrt{10} + 2 \cdot 5^{\frac{2}{2}} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$\frac{\sqrt{10} + 2 \cdot 5^{\frac{2}{2}} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{10} + 2 \cdot 5^{1} - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.4.5

Evaluate the exponent.

$\frac{\sqrt{10} + 2 \cdot 5 - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.5

Multiply $2$ by $5$ .

$\frac{\sqrt{10} + 10 - 2 \sqrt{2} \sqrt{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.6

Multiply $- 2 \sqrt{2} \sqrt{2}$ .

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Step 4.1.6.1

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

$\frac{\sqrt{10} + 10 - 2 ({\sqrt{2}}^{1} \sqrt{2}) - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.6.2

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

$\frac{\sqrt{10} + 10 - 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.6.3

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

$\frac{\sqrt{10} + 10 - 2 {\sqrt{2}}^{1 + 1} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.6.4

Add $1$ and $1$ .

$\frac{\sqrt{10} + 10 - 2 {\sqrt{2}}^{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7

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

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Step 4.1.7.1

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

$\frac{\sqrt{10} + 10 - 2 {(2^{\frac{1}{2}})}^{2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7.2

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

$\frac{\sqrt{10} + 10 - 2 \cdot 2^{\frac{1}{2} \cdot 2} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7.3

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

$\frac{\sqrt{10} + 10 - 2 \cdot 2^{\frac{2}{2}} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$\frac{\sqrt{10} + 10 - 2 \cdot 2^{\frac{2}{2}} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{\sqrt{10} + 10 - 2 \cdot 2^{1} - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.7.5

Evaluate the exponent.

$\frac{\sqrt{10} + 10 - 2 \cdot 2 - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.8

Multiply $- 2$ by $2$ .

$\frac{\sqrt{10} + 10 - 4 - 2 \sqrt{2} (2 \sqrt{5})}{- 18}$

Step 4.1.9

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

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Step 4.1.9.1

Multiply $2$ by $- 2$ .

$\frac{\sqrt{10} + 10 - 4 - 4 \sqrt{2} \sqrt{5}}{- 18}$

Step 4.1.9.2

Combine using the product rule for radicals.

$\frac{\sqrt{10} + 10 - 4 - 4 \sqrt{5 \cdot 2}}{- 18}$

Step 4.1.9.3

Multiply $5$ by $2$ .

$\frac{\sqrt{10} + 10 - 4 - 4 \sqrt{10}}{- 18}$

Step 4.2

Subtract $4 \sqrt{10}$ from $\sqrt{10}$ .

$\frac{10 - 4 - 3 \sqrt{10}}{- 18}$

Step 4.3

Subtract $4$ from $10$ .

$\frac{6 - 3 \sqrt{10}}{- 18}$

Step 5

Cancel the common factor of $6 - 3 \sqrt{10}$ and $- 18$ .

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Step 5.1

Factor $3$ out of $6$ .

$\frac{3 (2) - 3 \sqrt{10}}{- 18}$

Step 5.2

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

$\frac{3 (2) + 3 (- \sqrt{10})}{- 18}$

Step 5.3

Factor $3$ out of $3 (2) + 3 (- \sqrt{10})$ .

$\frac{3 (2 - \sqrt{10})}{- 18}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $3$ out of $- 18$ .

$\frac{3 (2 - \sqrt{10})}{3 \cdot - 6}$

Step 5.4.2

Cancel the common factor.

$\frac{3 (2 - \sqrt{10})}{3 \cdot - 6}$

Step 5.4.3

Rewrite the expression.

$\frac{2 - \sqrt{10}}{- 6}$

Step 6

Move the negative in front of the fraction.

$- \frac{2 - \sqrt{10}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 - \sqrt{10}}{6}$

Decimal Form:

$0.19371294 \dots$