Basic Math Examples

$(\sqrt{5} - \sqrt{13}) (\sqrt{5} + 13)$

Step 1

Expand $(\sqrt{5} - \sqrt{13}) (\sqrt{5} + 13)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{5} (\sqrt{5} + 13) - \sqrt{13} (\sqrt{5} + 13)$

Step 1.2

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} \cdot 13 - \sqrt{13} (\sqrt{5} + 13)$

Step 1.3

Apply the distributive property.

$\sqrt{5} \sqrt{5} + \sqrt{5} \cdot 13 - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2

Simplify each term.

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Step 2.1

Combine using the product rule for radicals.

$\sqrt{5 \cdot 5} + \sqrt{5} \cdot 13 - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2.2

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} \cdot 13 - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2.3

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

$\sqrt{5^{2}} + \sqrt{5} \cdot 13 - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2.4

Pull terms out from under the radical, assuming positive real numbers.

$5 + \sqrt{5} \cdot 13 - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2.5

Move $13$ to the left of $\sqrt{5}$ .

$5 + 13 \cdot \sqrt{5} - \sqrt{13} \sqrt{5} - \sqrt{13} \cdot 13$

Step 2.6

Multiply $- \sqrt{13} \sqrt{5}$ .

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Step 2.6.1

Combine using the product rule for radicals.

$5 + 13 \sqrt{5} - \sqrt{5 \cdot 13} - \sqrt{13} \cdot 13$

Step 2.6.2

Multiply $5$ by $13$ .

$5 + 13 \sqrt{5} - \sqrt{65} - \sqrt{13} \cdot 13$

Step 2.7

Multiply $13$ by $- 1$ .

$5 + 13 \sqrt{5} - \sqrt{65} - 13 \sqrt{13}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5 + 13 \sqrt{5} - \sqrt{65} - 13 \sqrt{13}$

Decimal Form:

$- 20.86554062 \dots$