Trigonometry Examples

$5 \sqrt{108 + 10 \sqrt{48}}$

Step 1

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.1

Factor $16$ out of $48$ .

$5 \sqrt{108 + 10 \sqrt{16 (3)}}$

Step 1.2

Rewrite $16$ as $4^{2}$ .

$5 \sqrt{108 + 10 \sqrt{4^{2} \cdot 3}}$

$5 \sqrt{108 + 10 \sqrt{4^{2} \cdot 3}}$

Step 2

Pull terms out from under the radical.

$5 \sqrt{108 + 10 (4 \sqrt{3})}$

Step 3

Multiply $4$ by $10$ .

$5 \sqrt{108 + 40 \sqrt{3}}$

Step 4

Rewrite $108 + 40 \sqrt{3}$ as $2^{2} (27 + 10 \sqrt{3})$ .

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Step 4.1

Factor $4$ out of $108$ .

$5 \sqrt{4 (27) + 40 \sqrt{3}}$

Step 4.2

Factor $4$ out of $40 \sqrt{3}$ .

$5 \sqrt{4 (27) + 4 (10 \sqrt{3})}$

Step 4.3

Factor $4$ out of $4 (27) + 4 (10 \sqrt{3})$ .

$5 \sqrt{4 (27 + 10 \sqrt{3})}$

Step 4.4

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

$5 \sqrt{2^{2} (27 + 10 \sqrt{3})}$

$5 \sqrt{2^{2} (27 + 10 \sqrt{3})}$

Step 5

Pull terms out from under the radical.

$5 (2 \sqrt{27 + 10 \sqrt{3}})$

Step 6

Multiply $2$ by $5$ .

$10 \sqrt{27 + 10 \sqrt{3}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{27 + 10 \sqrt{3}}$

Decimal Form:

$66.57364949 \dots$

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