Trigonometry Examples

$\sqrt{10} (4 \sqrt{2} + \sqrt{5})$

Step 1

Apply the distributive property.

$\sqrt{10} (4 \sqrt{2}) + \sqrt{10} \sqrt{5}$

Step 2

Multiply $\sqrt{10} (4 \sqrt{2})$ .

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Step 2.1

Combine using the product rule for radicals.

$4 \sqrt{10 \cdot 2} + \sqrt{10} \sqrt{5}$

Step 2.2

Multiply $10$ by $2$ .

$4 \sqrt{20} + \sqrt{10} \sqrt{5}$

$4 \sqrt{20} + \sqrt{10} \sqrt{5}$

Step 3

Combine using the product rule for radicals.

$4 \sqrt{20} + \sqrt{10 \cdot 5}$

Step 4

Simplify each term.

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Step 4.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 4.1.1

Factor $4$ out of $20$ .

$4 \sqrt{4 (5)} + \sqrt{10 \cdot 5}$

Step 4.1.2

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

$4 \sqrt{2^{2} \cdot 5} + \sqrt{10 \cdot 5}$

$4 \sqrt{2^{2} \cdot 5} + \sqrt{10 \cdot 5}$

Step 4.2

Pull terms out from under the radical.

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

Step 4.3

Multiply $2$ by $4$ .

$8 \sqrt{5} + \sqrt{10 \cdot 5}$

Step 4.4

Multiply $10$ by $5$ .

$8 \sqrt{5} + \sqrt{50}$

Step 4.5

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

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Step 4.5.1

Factor $25$ out of $50$ .

$8 \sqrt{5} + \sqrt{25 (2)}$

Step 4.5.2

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

$8 \sqrt{5} + \sqrt{5^{2} \cdot 2}$

$8 \sqrt{5} + \sqrt{5^{2} \cdot 2}$

Step 4.6

Pull terms out from under the radical.

$8 \sqrt{5} + 5 \sqrt{2}$

$8 \sqrt{5} + 5 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{5} + 5 \sqrt{2}$

Decimal Form:

$24.95961163 \dots$

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