Calculus Examples

$(5 - 3 \sqrt{6}) (2 - 4 \sqrt{10})$

Step 1

Expand $(5 - 3 \sqrt{6}) (2 - 4 \sqrt{10})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$5 (2 - 4 \sqrt{10}) - 3 \sqrt{6} (2 - 4 \sqrt{10})$

Step 1.2

Apply the distributive property.

$5 \cdot 2 + 5 (- 4 \sqrt{10}) - 3 \sqrt{6} (2 - 4 \sqrt{10})$

Step 1.3

Apply the distributive property.

$5 \cdot 2 + 5 (- 4 \sqrt{10}) - 3 \sqrt{6} \cdot 2 - 3 \sqrt{6} (- 4 \sqrt{10})$

Step 2

Simplify each term.

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Step 2.1

Multiply $5$ by $2$ .

$10 + 5 (- 4 \sqrt{10}) - 3 \sqrt{6} \cdot 2 - 3 \sqrt{6} (- 4 \sqrt{10})$

Step 2.2

Multiply $- 4$ by $5$ .

$10 - 20 \sqrt{10} - 3 \sqrt{6} \cdot 2 - 3 \sqrt{6} (- 4 \sqrt{10})$

Step 2.3

Multiply $2$ by $- 3$ .

$10 - 20 \sqrt{10} - 6 \sqrt{6} - 3 \sqrt{6} (- 4 \sqrt{10})$

Step 2.4

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

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Step 2.4.1

Multiply $- 4$ by $- 3$ .

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 \sqrt{6} \sqrt{10}$

Step 2.4.2

Combine using the product rule for radicals.

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 \sqrt{10 \cdot 6}$

Step 2.4.3

Multiply $10$ by $6$ .

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 \sqrt{60}$

Step 2.5

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

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Step 2.5.1

Factor $4$ out of $60$ .

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 \sqrt{4 (15)}$

Step 2.5.2

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

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 \sqrt{2^{2} \cdot 15}$

Step 2.6

Pull terms out from under the radical.

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 12 (2 \sqrt{15})$

Step 2.7

Multiply $2$ by $12$ .

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 24 \sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$10 - 20 \sqrt{10} - 6 \sqrt{6} + 24 \sqrt{15}$

Decimal Form:

$25.00910864 \dots$