Trigonometry Examples

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

Step 1

Apply the product rule to $2 \sqrt{3}$ .

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

Step 2

Multiply $2^{2}$ by $2^{2}$ by adding the exponents.

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Step 2.1

Move $2^{2}$ .

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

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3

Add $2$ and $2$ .

$\sqrt{2^{4} {\sqrt{3}}^{2}}$

Step 3

Raise $2$ to the power of $4$ .

$\sqrt{16 {\sqrt{3}}^{2}}$

Step 4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{16 {(3^{\frac{1}{2}})}^{2}}$

Step 4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{16 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 4.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{16 \cdot 3^{\frac{2}{2}}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\sqrt{16 \cdot 3^{\frac{2}{2}}}$

Step 4.4.2

Rewrite the expression.

$\sqrt{16 \cdot 3^{1}}$

Step 4.5

Evaluate the exponent.

$\sqrt{16 \cdot 3}$

Step 5

Multiply $16$ by $3$ .

$\sqrt{48}$

Step 6

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

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Step 6.1

Factor $16$ out of $48$ .

$\sqrt{16 (3)}$

Step 6.2

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

$\sqrt{4^{2} \cdot 3}$

Step 7

Pull terms out from under the radical.

$4 \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{3}$

Decimal Form:

$6.92820323 \dots$