Algebra Examples

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{3} - \sqrt{10}}$

Step 1

Multiply $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{3} - \sqrt{10}}$ by $\frac{\sqrt{3} + \sqrt{10}}{\sqrt{3} + \sqrt{10}}$ .

$\frac{\sqrt{5} - \sqrt{6}}{\sqrt{3} - \sqrt{10}} \cdot \frac{\sqrt{3} + \sqrt{10}}{\sqrt{3} + \sqrt{10}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{5} - \sqrt{6}}{\sqrt{3} - \sqrt{10}}$ by $\frac{\sqrt{3} + \sqrt{10}}{\sqrt{3} + \sqrt{10}}$ .

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{3} + \sqrt{10})}{(\sqrt{3} - \sqrt{10}) (\sqrt{3} + \sqrt{10})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{3} + \sqrt{10})}{{\sqrt{3}}^{2} + \sqrt{30} - \sqrt{30} - {\sqrt{10}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} - \sqrt{6}) (\sqrt{3} + \sqrt{10})}{- 7}$

Step 3

Expand $(\sqrt{5} - \sqrt{6}) (\sqrt{3} + \sqrt{10})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{3} + \sqrt{10}) - \sqrt{6} (\sqrt{3} + \sqrt{10})}{- 7}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{3} + \sqrt{5} \sqrt{10} - \sqrt{6} (\sqrt{3} + \sqrt{10})}{- 7}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{3} + \sqrt{5} \sqrt{10} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

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.

$\frac{\sqrt{5 \cdot 3} + \sqrt{5} \sqrt{10} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.2

Multiply $5$ by $3$ .

$\frac{\sqrt{15} + \sqrt{5} \sqrt{10} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{\sqrt{15} + \sqrt{5 \cdot 10} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.4

Multiply $5$ by $10$ .

$\frac{\sqrt{15} + \sqrt{50} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.5

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

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Step 4.1.5.1

Factor $25$ out of $50$ .

$\frac{\sqrt{15} + \sqrt{25 (2)} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.5.2

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

$\frac{\sqrt{15} + \sqrt{5^{2} \cdot 2} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.6

Pull terms out from under the radical.

$\frac{\sqrt{15} + 5 \sqrt{2} - \sqrt{6} \sqrt{3} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.7

Multiply $- \sqrt{6} \sqrt{3}$ .

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Step 4.1.7.1

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 5 \sqrt{2} - \sqrt{3 \cdot 6} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.7.2

Multiply $3$ by $6$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - \sqrt{18} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.8

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.8.1

Factor $9$ out of $18$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - \sqrt{9 (2)} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.8.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - \sqrt{3^{2} \cdot 2} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.9

Pull terms out from under the radical.

$\frac{\sqrt{15} + 5 \sqrt{2} - (3 \sqrt{2}) - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.10

Multiply $3$ by $- 1$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - \sqrt{6} \sqrt{10}}{- 7}$

Step 4.1.11

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

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Step 4.1.11.1

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - \sqrt{10 \cdot 6}}{- 7}$

Step 4.1.11.2

Multiply $10$ by $6$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - \sqrt{60}}{- 7}$

Step 4.1.12

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

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Step 4.1.12.1

Factor $4$ out of $60$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - \sqrt{4 (15)}}{- 7}$

Step 4.1.12.2

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

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - \sqrt{2^{2} \cdot 15}}{- 7}$

Step 4.1.13

Pull terms out from under the radical.

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - (2 \sqrt{15})}{- 7}$

Step 4.1.14

Multiply $2$ by $- 1$ .

$\frac{\sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2} - 2 \sqrt{15}}{- 7}$

Step 4.2

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

$\frac{- \sqrt{15} + 5 \sqrt{2} - 3 \sqrt{2}}{- 7}$

Step 4.3

Subtract $3 \sqrt{2}$ from $5 \sqrt{2}$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

Factor $- 1$ out of $- \sqrt{15}$ .

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

Step 7

Factor $- 1$ out of $2 \sqrt{2}$ .

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

Step 8

Factor $- 1$ out of $- (\sqrt{15}) - (- 2 \sqrt{2})$ .

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

Step 9

Simplify the expression.

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Step 9.1

Rewrite $- (\sqrt{15} - 2 \sqrt{2})$ as $- 1 (\sqrt{15} - 2 \sqrt{2})$ .

$- \frac{- 1 (\sqrt{15} - 2 \sqrt{2})}{7}$

Step 9.2

Move the negative in front of the fraction.

$- - \frac{\sqrt{15} - 2 \sqrt{2}}{7}$

Step 9.3

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{15} - 2 \sqrt{2}}{7}$

Step 9.4

Multiply $\frac{\sqrt{15} - 2 \sqrt{2}}{7}$ by $1$ .

$\frac{\sqrt{15} - 2 \sqrt{2}}{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15} - 2 \sqrt{2}}{7}$

Decimal Form:

$0.14922231 \dots$