Simplify ( square root of 5- square root of 6)( square root of 5+ square root of 2)

$(\sqrt{5} - \sqrt{6}) (\sqrt{5} + \sqrt{2})$

Step 1

Expand $(\sqrt{5} - \sqrt{6}) (\sqrt{5} + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{5} (\sqrt{5} + \sqrt{2}) - \sqrt{6} (\sqrt{5} + \sqrt{2})$

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{2} - \sqrt{6} (\sqrt{5} + \sqrt{2})$

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Step 2

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{5 \cdot 5} + \sqrt{5} \sqrt{2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} \sqrt{2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

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

$\sqrt{5^{2}} + \sqrt{5} \sqrt{2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Pull terms out from under the radical, assuming positive real numbers.

$5 + \sqrt{5} \sqrt{2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Combine using the product rule for radicals.

$5 + \sqrt{5 \cdot 2} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Multiply $5$ by $2$ .

$5 + \sqrt{10} - \sqrt{6} \sqrt{5} - \sqrt{6} \sqrt{2}$

Multiply $- \sqrt{6} \sqrt{5}$ .

Combine using the product rule for radicals.

$5 + \sqrt{10} - \sqrt{5 \cdot 6} - \sqrt{6} \sqrt{2}$

Multiply $5$ by $6$ .

$5 + \sqrt{10} - \sqrt{30} - \sqrt{6} \sqrt{2}$

Multiply $- \sqrt{6} \sqrt{2}$ .

Combine using the product rule for radicals.

$5 + \sqrt{10} - \sqrt{30} - \sqrt{2 \cdot 6}$

Multiply $2$ by $6$ .

$5 + \sqrt{10} - \sqrt{30} - \sqrt{12}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$5 + \sqrt{10} - \sqrt{30} - \sqrt{4 (3)}$

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

$5 + \sqrt{10} - \sqrt{30} - \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

$5 + \sqrt{10} - \sqrt{30} - 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5 + \sqrt{10} - \sqrt{30} - 2 \sqrt{3}$

Decimal Form:

$- 0.77904953 \dots$

Simplify ( square root of 5- square root of 6)( square root of 5+ square root of 2)

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