Simplify ( square root of 12+ square root of 6)( square root of 6- square root of 10)

$(\sqrt{12} + \sqrt{6}) (\sqrt{6} - \sqrt{10})$

Step 1

Simplify each term.

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

Factor $4$ out of $12$ .

$(\sqrt{4 (3)} + \sqrt{6}) (\sqrt{6} - \sqrt{10})$

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

$(\sqrt{2^{2} \cdot 3} + \sqrt{6}) (\sqrt{6} - \sqrt{10})$

Pull terms out from under the radical.

$(2 \sqrt{3} + \sqrt{6}) (\sqrt{6} - \sqrt{10})$

Step 2

Expand $(2 \sqrt{3} + \sqrt{6}) (\sqrt{6} - \sqrt{10})$ using the FOIL Method.

Apply the distributive property.

$2 \sqrt{3} (\sqrt{6} - \sqrt{10}) + \sqrt{6} (\sqrt{6} - \sqrt{10})$

Apply the distributive property.

$2 \sqrt{3} \sqrt{6} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} (\sqrt{6} - \sqrt{10})$

Apply the distributive property.

$2 \sqrt{3} \sqrt{6} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Step 3

Simplify each term.

Multiply $2 \sqrt{3} \sqrt{6}$ .

Combine using the product rule for radicals.

$2 \sqrt{6 \cdot 3} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Multiply $6$ by $3$ .

$2 \sqrt{18} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

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

Factor $9$ out of $18$ .

$2 \sqrt{9 (2)} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

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

$2 \sqrt{3^{2} \cdot 2} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Pull terms out from under the radical.

$2 (3 \sqrt{2}) + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Multiply $3$ by $2$ .

$6 \sqrt{2} + 2 \sqrt{3} (- \sqrt{10}) + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Multiply $2 \sqrt{3} (- \sqrt{10})$ .

Multiply $- 1$ by $2$ .

$6 \sqrt{2} - 2 \sqrt{3} \sqrt{10} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Combine using the product rule for radicals.

$6 \sqrt{2} - 2 \sqrt{10 \cdot 3} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Multiply $10$ by $3$ .

$6 \sqrt{2} - 2 \sqrt{30} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{10})$

Combine using the product rule for radicals.

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

Multiply $6$ by $6$ .

$6 \sqrt{2} - 2 \sqrt{30} + \sqrt{36} + \sqrt{6} (- \sqrt{10})$

Rewrite $36$ as $6^{2}$ .

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

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

$6 \sqrt{2} - 2 \sqrt{30} + 6 + \sqrt{6} (- \sqrt{10})$

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

Combine using the product rule for radicals.

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

Multiply $6$ by $10$ .

$6 \sqrt{2} - 2 \sqrt{30} + 6 - \sqrt{60}$

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

Factor $4$ out of $60$ .

$6 \sqrt{2} - 2 \sqrt{30} + 6 - \sqrt{4 (15)}$

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

$6 \sqrt{2} - 2 \sqrt{30} + 6 - \sqrt{2^{2} \cdot 15}$

Pull terms out from under the radical.

$6 \sqrt{2} - 2 \sqrt{30} + 6 - (2 \sqrt{15})$

Multiply $2$ by $- 1$ .

$6 \sqrt{2} - 2 \sqrt{30} + 6 - 2 \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{2} - 2 \sqrt{30} + 6 - 2 \sqrt{15}$

Decimal Form:

$- 4.21513646 \dots$

Simplify ( square root of 12+ square root of 6)( square root of 6- square root of 10)

Related Questions

Support LCMGCF.com by disabling your adblocker.