Finite Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Combine using the product rule for radicals.

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

Step 3.2.1.1.2

Multiply $5$ by $2$ .

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

Multiply $- 1$ by $2$ .

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

Step 3.2.1.2.2

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

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

Step 3.2.1.2.3

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

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

Step 3.2.1.2.4

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

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

Step 3.2.1.2.5

Add $1$ and $1$ .

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

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

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

Step 3.2.1.3.3

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

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

Step 3.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.2.1.3.4.1

Cancel the common factor.

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

Step 3.2.1.3.4.2

Rewrite the expression.

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

Step 3.2.1.3.5

Evaluate the exponent.

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

Step 3.2.1.4

Multiply $- 2$ by $2$ .

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

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

$\frac{2 \sqrt{10} - 4 + 2 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.5.2

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

$\frac{2 \sqrt{10} - 4 + 2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.5.3

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

$\frac{2 \sqrt{10} - 4 + 2 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.5.4

Add $1$ and $1$ .

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

Step 3.2.1.6

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

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Step 3.2.1.6.1

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

$\frac{2 \sqrt{10} - 4 + 2 {(5^{\frac{1}{2}})}^{2} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.6.2

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

$\frac{2 \sqrt{10} - 4 + 2 \cdot 5^{\frac{1}{2} \cdot 2} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.6.3

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

$\frac{2 \sqrt{10} - 4 + 2 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.6.4

Cancel the common factor of $2$ .

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Step 3.2.1.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{10} - 4 + 2 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{10} - 4 + 2 \cdot 5^{1} + 2 \sqrt{5} (- \sqrt{2})}{3}$

Step 3.2.1.6.5

Evaluate the exponent.

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

Step 3.2.1.7

Multiply $2$ by $5$ .

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

Step 3.2.1.8

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

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Step 3.2.1.8.1

Multiply $- 1$ by $2$ .

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

Step 3.2.1.8.2

Combine using the product rule for radicals.

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

Step 3.2.1.8.3

Multiply $2$ by $5$ .

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

Step 3.2.2

Subtract $2 \sqrt{10}$ from $2 \sqrt{10}$ .

$\frac{0 - 4 + 10}{3}$

Step 3.2.3

Subtract $4$ from $0$ .

$\frac{- 4 + 10}{3}$

Step 3.2.4

Add $- 4$ and $10$ .

$\frac{6}{3}$

Step 4

Divide $6$ by $3$ .

$2$