Trigonometry Examples

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

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

Step 1

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

\sqrt{{(3^{\frac{1}{2}})}^{2} + {(\sqrt{3})}^{2}}

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

Step 1.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{3^{\frac{1}{2} \cdot 2} + {(\sqrt{3})}^{2}}

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

Step 1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 1.4

Cancel the common factor of

2

$2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$\sqrt{3^{1} + {(\sqrt{3})}^{2}}$

$\sqrt{3^{1} + {(\sqrt{3})}^{2}}$

Step 1.5

Evaluate the exponent.

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

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

Step 2

Rewrite

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

$3$ .

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Step 2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

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

Step 2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 2.4

Cancel the common factor of

$2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$\sqrt{3 + 3^{1}}$

$\sqrt{3 + 3^{1}}$

Step 2.5

Evaluate the exponent.

$\sqrt{3 + 3}$

$\sqrt{3 + 3}$

Step 3

Add

$3$ and

$3$ .

$\sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{6}$

Decimal Form:

$2.44948974 \dots$

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