Algebra Examples

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

Step 1

Multiply $\frac{\sqrt{10} - \sqrt{5}}{\sqrt{5} + \sqrt{2}}$ by $\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{10} - \sqrt{5}}{\sqrt{5} + \sqrt{2}}$ by $\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Expand $(\sqrt{10} - \sqrt{5}) (\sqrt{5} - \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify each term.

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

Multiply $10$ by $5$ .

$\frac{\sqrt{50} + \sqrt{10} (- \sqrt{2}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- \sqrt{2})}{3}$

Step 4.3

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 4.3.1

Factor $25$ out of $50$ .

$\frac{\sqrt{25 (2)} + \sqrt{10} (- \sqrt{2}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- \sqrt{2})}{3}$

Step 4.3.2

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

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

Step 4.4

Pull terms out from under the radical.

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

Step 4.5

Multiply $\sqrt{10} (- \sqrt{2})$ .

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Step 4.5.1

Combine using the product rule for radicals.

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

Step 4.5.2

Multiply $10$ by $2$ .

$\frac{5 \sqrt{2} - \sqrt{20} - \sqrt{5} \sqrt{5} - \sqrt{5} (- \sqrt{2})}{3}$

Step 4.6

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 4.6.1

Factor $4$ out of $20$ .

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

Step 4.6.2

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

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

Step 4.7

Pull terms out from under the radical.

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

Step 4.8

Multiply $2$ by $- 1$ .

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

Step 4.9

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

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Step 4.9.1

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

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

Step 4.9.2

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

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

Step 4.9.3

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

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

Step 4.9.4

Add $1$ and $1$ .

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

Step 4.10

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

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Step 4.10.1

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

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

Step 4.10.2

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

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

Step 4.10.3

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

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

Step 4.10.4

Cancel the common factor of $2$ .

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Step 4.10.4.1

Cancel the common factor.

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

Step 4.10.4.2

Rewrite the expression.

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

Step 4.10.5

Evaluate the exponent.

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

Step 4.11

Multiply $- 1$ by $5$ .

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

Step 4.12

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

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Step 4.12.1

Multiply $- 1$ by $- 1$ .

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

Step 4.12.2

Multiply $\sqrt{5}$ by $1$ .

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

Step 4.12.3

Combine using the product rule for radicals.

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

Step 4.12.4

Multiply $5$ by $2$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25373650 \dots$