Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2

Combine using the product rule for radicals.

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

Step 2.1.3

Multiply $5$ by $10$ .

$(3 \sqrt{10} - 3 \sqrt{2} + \sqrt{50} + \sqrt{5} (- \sqrt{2})) \sqrt{3 - \sqrt{5}}$

Step 2.1.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.1.4.1

Factor $25$ out of $50$ .

$(3 \sqrt{10} - 3 \sqrt{2} + \sqrt{25 (2)} + \sqrt{5} (- \sqrt{2})) \sqrt{3 - \sqrt{5}}$

Step 2.1.4.2

Rewrite $25$ as $5^{2}$ .

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

Step 2.1.5

Pull terms out from under the radical.

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

Step 2.1.6

Multiply $\sqrt{5} (- \sqrt{2})$ .

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Step 2.1.6.1

Combine using the product rule for radicals.

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

Step 2.1.6.2

Multiply $5$ by $2$ .

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

Step 2.2

Subtract $\sqrt{10}$ from $3 \sqrt{10}$ .

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

Step 2.3

Add $- 3 \sqrt{2}$ and $5 \sqrt{2}$ .

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

Step 3

Apply the distributive property.

$2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 2 \sqrt{2} \sqrt{3 - \sqrt{5}}$

Step 4

Combine using the product rule for radicals.

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

Step 5

Combine using the product rule for radicals.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$8$