Algebra Examples

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

Step 1

Expand $(\sqrt{2} + \sqrt{3}) (\sqrt{5} - \sqrt{7})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Combine using the product rule for radicals.

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

Step 2.2

Multiply $2$ by $5$ .

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

Step 2.3

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

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Step 2.3.1

Combine using the product rule for radicals.

$\sqrt{10} - \sqrt{2 \cdot 7} + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{7})$

Step 2.3.2

Multiply $2$ by $7$ .

$\sqrt{10} - \sqrt{14} + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{7})$

Step 2.4

Combine using the product rule for radicals.

$\sqrt{10} - \sqrt{14} + \sqrt{3 \cdot 5} + \sqrt{3} (- \sqrt{7})$

Step 2.5

Multiply $3$ by $5$ .

$\sqrt{10} - \sqrt{14} + \sqrt{15} + \sqrt{3} (- \sqrt{7})$

Step 2.6

Multiply $\sqrt{3} (- \sqrt{7})$ .

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Step 2.6.1

Combine using the product rule for radicals.

$\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{3 \cdot 7}$

Step 2.6.2

Multiply $3$ by $7$ .

$\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$

Decimal Form:

$- 1.28897207 \dots$