Trigonometry Examples

$\sqrt{2^{3}} \sqrt{2 + \sqrt{18}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{2^{3} (2 + \sqrt{18})}$

Step 2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1

Factor $9$ out of $18$ .

$\sqrt{2^{3} (2 + \sqrt{9 (2)})}$

Step 2.2

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

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

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

Step 3

Pull terms out from under the radical.

$\sqrt{2^{3} (2 + 3 \sqrt{2})}$

Step 4

Raise $2$ to the power of $3$ .

$\sqrt{8 (2 + 3 \sqrt{2})}$

Step 5

Rewrite $8 (2 + 3 \sqrt{2})$ as $2^{2} \cdot (2 (2 + 3 \sqrt{2}))$ .

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Step 5.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) (2 + 3 \sqrt{2})}$

Step 5.2

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

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

Step 5.3

Add parentheses.

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

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

Step 6

Pull terms out from under the radical.

$2 \sqrt{2 (2 + 3 \sqrt{2})}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2 (2 + 3 \sqrt{2})}$

Decimal Form:

$7.06690352 \dots$

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