Pre-Algebra Examples

$\frac{\sqrt{10} - \sqrt{7}}{\sqrt{6} - \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{10} - \sqrt{7}}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

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

Step 2

Multiply $\frac{\sqrt{10} - \sqrt{7}}{\sqrt{6} - \sqrt{2}}$ by $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{10} - \sqrt{7}) (\sqrt{6} + \sqrt{2})}{{\sqrt{6}}^{2} + \sqrt{12} - \sqrt{12} - {\sqrt{2}}^{2}}$

Step 4

Simplify.

$\frac{(\sqrt{10} - \sqrt{7}) (\sqrt{6} + \sqrt{2})}{4}$

Step 5

Expand $(\sqrt{10} - \sqrt{7}) (\sqrt{6} + \sqrt{2})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{10} (\sqrt{6} + \sqrt{2}) - \sqrt{7} (\sqrt{6} + \sqrt{2})}{4}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{6} + \sqrt{10} \sqrt{2} - \sqrt{7} (\sqrt{6} + \sqrt{2})}{4}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{6} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6

Simplify each term.

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 6} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.2

Multiply $10$ by $6$ .

$\frac{\sqrt{60} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.3

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

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Step 6.3.1

Factor $4$ out of $60$ .

$\frac{\sqrt{4 (15)} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.3.2

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

$\frac{\sqrt{2^{2} \cdot 15} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.4

Pull terms out from under the radical.

$\frac{2 \sqrt{15} + \sqrt{10} \sqrt{2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.5

Combine using the product rule for radicals.

$\frac{2 \sqrt{15} + \sqrt{10 \cdot 2} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.6

Multiply $10$ by $2$ .

$\frac{2 \sqrt{15} + \sqrt{20} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.7

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

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Step 6.7.1

Factor $4$ out of $20$ .

$\frac{2 \sqrt{15} + \sqrt{4 (5)} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.7.2

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

$\frac{2 \sqrt{15} + \sqrt{2^{2} \cdot 5} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.8

Pull terms out from under the radical.

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{7} \sqrt{6} - \sqrt{7} \sqrt{2}}{4}$

Step 6.9

Multiply $- \sqrt{7} \sqrt{6}$ .

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Step 6.9.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{6 \cdot 7} - \sqrt{7} \sqrt{2}}{4}$

Step 6.9.2

Multiply $6$ by $7$ .

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42} - \sqrt{7} \sqrt{2}}{4}$

Step 6.10

Multiply $- \sqrt{7} \sqrt{2}$ .

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Step 6.10.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42} - \sqrt{2 \cdot 7}}{4}$

Step 6.10.2

Multiply $2$ by $7$ .

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42} - \sqrt{14}}{4}$

Step 7

Split the fraction $\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42} - \sqrt{14}}{4}$ into two fractions.

$\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 8

Split the fraction $\frac{2 \sqrt{15} + 2 \sqrt{5} - \sqrt{42}}{4}$ into two fractions.

$\frac{2 \sqrt{15} + 2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 9

Split the fraction $\frac{2 \sqrt{15} + 2 \sqrt{5}}{4}$ into two fractions.

$\frac{2 \sqrt{15}}{4} + \frac{2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 10

Cancel the common factor of $2$ and $4$ .

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Step 10.1

Factor $2$ out of $2 \sqrt{15}$ .

$\frac{2 (\sqrt{15})}{4} + \frac{2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt{15}}{2 \cdot 2} + \frac{2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 10.2.2

Cancel the common factor.

$\frac{2 \sqrt{15}}{2 \cdot 2} + \frac{2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 10.2.3

Rewrite the expression.

$\frac{\sqrt{15}}{2} + \frac{2 \sqrt{5}}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 11

Cancel the common factor of $2$ and $4$ .

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Step 11.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{\sqrt{15}}{2} + \frac{2 (\sqrt{5})}{4} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $4$ .

$\frac{\sqrt{15}}{2} + \frac{2 \sqrt{5}}{2 \cdot 2} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 11.2.2

Cancel the common factor.

$\frac{\sqrt{15}}{2} + \frac{2 \sqrt{5}}{2 \cdot 2} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 11.2.3

Rewrite the expression.

$\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2} + \frac{- \sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 12

Move the negative in front of the fraction.

$\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2} - \frac{\sqrt{42}}{4} + \frac{- \sqrt{14}}{4}$

Step 13

Move the negative in front of the fraction.

$\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2} - \frac{\sqrt{42}}{4} - \frac{\sqrt{14}}{4}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15}}{2} + \frac{\sqrt{5}}{2} - \frac{\sqrt{42}}{4} - \frac{\sqrt{14}}{4}$

Decimal Form:

$0.49892614 \dots$