Basic Math Examples

$(\sqrt{3} - \sqrt{5}) (\sqrt{3} \sqrt{5})$

Step 1

Combine using the product rule for radicals.

$(\sqrt{3} - \sqrt{5}) \sqrt{3 \cdot 5}$

Step 2

Multiply $3$ by $5$ .

$(\sqrt{3} - \sqrt{5}) \sqrt{15}$

Step 3

Apply the distributive property.

$\sqrt{3} \sqrt{15} - \sqrt{5} \sqrt{15}$

Step 4

Combine using the product rule for radicals.

$\sqrt{3 \cdot 15} - \sqrt{5} \sqrt{15}$

Step 5

Multiply $- \sqrt{5} \sqrt{15}$ .

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Step 5.1

Combine using the product rule for radicals.

$\sqrt{3 \cdot 15} - \sqrt{15 \cdot 5}$

Step 5.2

Multiply $15$ by $5$ .

$\sqrt{3 \cdot 15} - \sqrt{75}$

$\sqrt{3 \cdot 15} - \sqrt{75}$

Step 6

Simplify each term.

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Step 6.1

Multiply $3$ by $15$ .

$\sqrt{45} - \sqrt{75}$

Step 6.2

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 6.2.1

Factor $9$ out of $45$ .

$\sqrt{9 (5)} - \sqrt{75}$

Step 6.2.2

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

$\sqrt{3^{2} \cdot 5} - \sqrt{75}$

$\sqrt{3^{2} \cdot 5} - \sqrt{75}$

Step 6.3

Pull terms out from under the radical.

$3 \sqrt{5} - \sqrt{75}$

Step 6.4

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 6.4.1

Factor $25$ out of $75$ .

$3 \sqrt{5} - \sqrt{25 (3)}$

Step 6.4.2

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

$3 \sqrt{5} - \sqrt{5^{2} \cdot 3}$

$3 \sqrt{5} - \sqrt{5^{2} \cdot 3}$

Step 6.5

Pull terms out from under the radical.

$3 \sqrt{5} - (5 \sqrt{3})$

Step 6.6

Multiply $5$ by $- 1$ .

$3 \sqrt{5} - 5 \sqrt{3}$

$3 \sqrt{5} - 5 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5} - 5 \sqrt{3}$

Decimal Form:

$- 1.95205010 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$