Algebra Examples

$\frac{\sqrt{21} - \sqrt{6}}{\sqrt{35} - \sqrt{10}}$

Step 1

Multiply $\frac{\sqrt{21} - \sqrt{6}}{\sqrt{35} - \sqrt{10}}$ by $\frac{\sqrt{35} + \sqrt{10}}{\sqrt{35} + \sqrt{10}}$ .

$\frac{\sqrt{21} - \sqrt{6}}{\sqrt{35} - \sqrt{10}} \cdot \frac{\sqrt{35} + \sqrt{10}}{\sqrt{35} + \sqrt{10}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{21} - \sqrt{6}}{\sqrt{35} - \sqrt{10}}$ by $\frac{\sqrt{35} + \sqrt{10}}{\sqrt{35} + \sqrt{10}}$ .

$\frac{(\sqrt{21} - \sqrt{6}) (\sqrt{35} + \sqrt{10})}{(\sqrt{35} - \sqrt{10}) (\sqrt{35} + \sqrt{10})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{21} - \sqrt{6}) (\sqrt{35} + \sqrt{10})}{{\sqrt{35}}^{2} + \sqrt{350} - \sqrt{350} - {\sqrt{10}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{21} - \sqrt{6}) (\sqrt{35} + \sqrt{10})}{25}$

Step 3

Simplify the numerator.

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Step 3.1

Expand $(\sqrt{21} - \sqrt{6}) (\sqrt{35} + \sqrt{10})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{\sqrt{21} (\sqrt{35} + \sqrt{10}) - \sqrt{6} (\sqrt{35} + \sqrt{10})}{25}$

Step 3.1.2

Apply the distributive property.

$\frac{\sqrt{21} \sqrt{35} + \sqrt{21} \sqrt{10} - \sqrt{6} (\sqrt{35} + \sqrt{10})}{25}$

Step 3.1.3

Apply the distributive property.

$\frac{\sqrt{21} \sqrt{35} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

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

Combine using the product rule for radicals.

$\frac{\sqrt{21 \cdot 35} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.2

Multiply $21$ by $35$ .

$\frac{\sqrt{735} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.3

Rewrite $735$ as $7^{2} \cdot 15$ .

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Step 3.2.1.3.1

Factor $49$ out of $735$ .

$\frac{\sqrt{49 (15)} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.3.2

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2} \cdot 15} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.4

Pull terms out from under the radical.

$\frac{7 \sqrt{15} + \sqrt{21} \sqrt{10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.5

Combine using the product rule for radicals.

$\frac{7 \sqrt{15} + \sqrt{21 \cdot 10} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.6

Multiply $21$ by $10$ .

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{6} \sqrt{35} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.7

Multiply $- \sqrt{6} \sqrt{35}$ .

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Step 3.2.1.7.1

Combine using the product rule for radicals.

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{35 \cdot 6} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.7.2

Multiply $35$ by $6$ .

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - \sqrt{6} \sqrt{10}}{25}$

Step 3.2.1.8

Multiply $- \sqrt{6} \sqrt{10}$ .

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Step 3.2.1.8.1

Combine using the product rule for radicals.

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - \sqrt{10 \cdot 6}}{25}$

Step 3.2.1.8.2

Multiply $10$ by $6$ .

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - \sqrt{60}}{25}$

Step 3.2.1.9

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 3.2.1.9.1

Factor $4$ out of $60$ .

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - \sqrt{4 (15)}}{25}$

Step 3.2.1.9.2

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

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - \sqrt{2^{2} \cdot 15}}{25}$

Step 3.2.1.10

Pull terms out from under the radical.

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - (2 \sqrt{15})}{25}$

Step 3.2.1.11

Multiply $2$ by $- 1$ .

$\frac{7 \sqrt{15} + \sqrt{210} - \sqrt{210} - 2 \sqrt{15}}{25}$

Step 3.2.2

Subtract $2 \sqrt{15}$ from $7 \sqrt{15}$ .

$\frac{5 \sqrt{15} + \sqrt{210} - \sqrt{210}}{25}$

Step 3.2.3

Subtract $\sqrt{210}$ from $\sqrt{210}$ .

$\frac{5 \sqrt{15} + 0}{25}$

Step 3.2.4

Add $5 \sqrt{15}$ and $0$ .

$\frac{5 \sqrt{15}}{25}$

Step 4

Cancel the common factor of $5$ and $25$ .

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Step 4.1

Factor $5$ out of $5 \sqrt{15}$ .

$\frac{5 (\sqrt{15})}{25}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $25$ .

$\frac{5 \sqrt{15}}{5 \cdot 5}$

Step 4.2.2

Cancel the common factor.

$\frac{5 \sqrt{15}}{5 \cdot 5}$

Step 4.2.3

Rewrite the expression.

$\frac{\sqrt{15}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15}}{5}$

Decimal Form:

$0.77459666 \dots$