Simplify the Radical Expression ( square root of 2+ square root of 3)( square root of 5- square root of 7)

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $2$ by $5$ .

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $7$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $5$ .

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $7$ .

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.28897207 \dots$