Simplify (7 square root of 2-3 square root of 3)(4 square root of 6+3 square root of 12)

$(7 \sqrt{2} - 3 \sqrt{3}) (4 \sqrt{6} + 3 \sqrt{12})$

Simplify each term.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $3$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $7 \sqrt{2} (4 \sqrt{6})$ .

Multiply $4$ by $7$ .

$28 \sqrt{2} \sqrt{6} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Combine using the product rule for radicals.

$28 \sqrt{6 \cdot 2} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $6$ by $2$ .

$28 \sqrt{12} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

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

Factor $4$ out of $12$ .

$28 \sqrt{4 (3)} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

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

$28 \sqrt{2^{2} \cdot 3} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Pull terms out from under the radical.

$28 (2 \sqrt{3}) + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $2$ by $28$ .

$56 \sqrt{3} + 7 \sqrt{2} (6 \sqrt{3}) - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $7 \sqrt{2} (6 \sqrt{3})$ .

Multiply $6$ by $7$ .

$56 \sqrt{3} + 42 \sqrt{2} \sqrt{3} - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Combine using the product rule for radicals.

$56 \sqrt{3} + 42 \sqrt{3 \cdot 2} - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $3$ by $2$ .

$56 \sqrt{3} + 42 \sqrt{6} - 3 \sqrt{3} (4 \sqrt{6}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $- 3 \sqrt{3} (4 \sqrt{6})$ .

Multiply $4$ by $- 3$ .

$56 \sqrt{3} + 42 \sqrt{6} - 12 \sqrt{3} \sqrt{6} - 3 \sqrt{3} (6 \sqrt{3})$

Combine using the product rule for radicals.

$56 \sqrt{3} + 42 \sqrt{6} - 12 \sqrt{6 \cdot 3} - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $6$ by $3$ .

$56 \sqrt{3} + 42 \sqrt{6} - 12 \sqrt{18} - 3 \sqrt{3} (6 \sqrt{3})$

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

Factor $9$ out of $18$ .

$56 \sqrt{3} + 42 \sqrt{6} - 12 \sqrt{9 (2)} - 3 \sqrt{3} (6 \sqrt{3})$

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

$56 \sqrt{3} + 42 \sqrt{6} - 12 \sqrt{3^{2} \cdot 2} - 3 \sqrt{3} (6 \sqrt{3})$

Pull terms out from under the radical.

$56 \sqrt{3} + 42 \sqrt{6} - 12 (3 \sqrt{2}) - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $3$ by $- 12$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 3 \sqrt{3} (6 \sqrt{3})$

Multiply $- 3 \sqrt{3} (6 \sqrt{3})$ .

Multiply $6$ by $- 3$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \sqrt{3} \sqrt{3}$

Raise $\sqrt{3}$ to the power of $1$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 ({\sqrt{3}}^{1} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 {\sqrt{3}}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 {(3^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \cdot 3^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \cdot 3^{1}$

Evaluate the exponent.

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 18 \cdot 3$

Multiply $- 18$ by $3$ .

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 54$

The result can be shown in multiple forms.

Exact Form:

$56 \sqrt{3} + 42 \sqrt{6} - 36 \sqrt{2} - 54$

Decimal Form:

$94.96172617 \dots$

Simplify (7 square root of 2-3 square root of 3)(4 square root of 6+3 square root of 12)

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