Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $3$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Multiply $4$ by $7$ .

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

Step 3.1.2

Combine using the product rule for radicals.

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

Step 3.1.3

Multiply $6$ by $2$ .

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

Step 3.2

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

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Step 3.2.1

Factor $4$ out of $12$ .

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

Step 3.2.2

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

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

Step 3.3

Pull terms out from under the radical.

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

Step 3.4

Multiply $2$ by $28$ .

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

Step 3.5

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

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Step 3.5.1

Multiply $6$ by $7$ .

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

Step 3.5.2

Combine using the product rule for radicals.

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

Step 3.5.3

Multiply $3$ by $2$ .

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

Step 3.6

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

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Step 3.6.1

Multiply $4$ by $- 3$ .

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

Step 3.6.2

Combine using the product rule for radicals.

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

Step 3.6.3

Multiply $6$ by $3$ .

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

Step 3.7

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

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Step 3.7.1

Factor $9$ out of $18$ .

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

Step 3.7.2

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

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

Step 3.8

Pull terms out from under the radical.

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

Step 3.9

Multiply $3$ by $- 12$ .

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

Step 3.10

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

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Step 3.10.1

Multiply $6$ by $- 3$ .

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

Step 3.10.2

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

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

Step 3.10.3

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

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

Step 3.10.4

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

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

Step 3.10.5

Add $1$ and $1$ .

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

Step 3.11

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

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Step 3.11.1

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

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

Step 3.11.2

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

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

Step 3.11.3

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

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

Step 3.11.4

Cancel the common factor of $2$ .

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Step 3.11.4.1

Cancel the common factor.

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

Step 3.11.4.2

Rewrite the expression.

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

Step 3.11.5

Evaluate the exponent.

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

Step 3.12

Multiply $- 18$ by $3$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$94.96172617 \dots$