Simplify (3+4 square root of 7)(6-2 square root of 14)

$(3 + 4 \sqrt{7}) (6 - 2 \sqrt{14})$

Expand $(3 + 4 \sqrt{7}) (6 - 2 \sqrt{14})$ using the FOIL Method.

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Apply the distributive property.

$3 (6 - 2 \sqrt{14}) + 4 \sqrt{7} (6 - 2 \sqrt{14})$

Apply the distributive property.

$3 \cdot 6 + 3 (- 2 \sqrt{14}) + 4 \sqrt{7} (6 - 2 \sqrt{14})$

Apply the distributive property.

$3 \cdot 6 + 3 (- 2 \sqrt{14}) + 4 \sqrt{7} \cdot 6 + 4 \sqrt{7} (- 2 \sqrt{14})$

Simplify each term.

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Multiply $3$ by $6$ .

$18 + 3 (- 2 \sqrt{14}) + 4 \sqrt{7} \cdot 6 + 4 \sqrt{7} (- 2 \sqrt{14})$

Multiply $- 2$ by $3$ .

$18 - 6 \sqrt{14} + 4 \sqrt{7} \cdot 6 + 4 \sqrt{7} (- 2 \sqrt{14})$

Multiply $6$ by $4$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} + 4 \sqrt{7} (- 2 \sqrt{14})$

Multiply $4 \sqrt{7} (- 2 \sqrt{14})$ .

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Multiply $- 2$ by $4$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 \sqrt{7} \sqrt{14}$

Combine using the product rule for radicals.

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 \sqrt{14 \cdot 7}$

Multiply $14$ by $7$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 \sqrt{98}$

Rewrite $98$ as $7^{2} \cdot 2$ .

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Factor $49$ out of $98$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 \sqrt{49 (2)}$

Rewrite $49$ as $7^{2}$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 \sqrt{7^{2} \cdot 2}$

Pull terms out from under the radical.

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 8 (7 \sqrt{2})$

Multiply $7$ by $- 8$ .

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 56 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$18 - 6 \sqrt{14} + 24 \sqrt{7} - 56 \sqrt{2}$

Decimal Form:

$- 20.14787234 \dots$

Simplify (3+4 square root of 7)(6-2 square root of 14)

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