Trigonometry Examples

\sqrt{{(\sqrt{6})}^{2} + {(- \sqrt{2})}^{2}}

$\sqrt{{(\sqrt{6})}^{2} + {(- \sqrt{2})}^{2}}$

Step 1

Rewrite

{\sqrt{6}}^{2}

${\sqrt{6}}^{2}$ as

6

$6$ .

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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{6}

$\sqrt{6}$ as

6^{\frac{1}{2}}

$6^{\frac{1}{2}}$ .

\sqrt{{(6^{\frac{1}{2}})}^{2} + {(- \sqrt{2})}^{2}}

$\sqrt{{(6^{\frac{1}{2}})}^{2} + {(- \sqrt{2})}^{2}}$

Step 1.2

Apply the power rule and multiply exponents,

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

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

\sqrt{6^{\frac{1}{2} \cdot 2} + {(- \sqrt{2})}^{2}}

$\sqrt{6^{\frac{1}{2} \cdot 2} + {(- \sqrt{2})}^{2}}$

Step 1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{6^{\frac{2}{2}} + {(- \sqrt{2})}^{2}}

$\sqrt{6^{\frac{2}{2}} + {(- \sqrt{2})}^{2}}$

Step 1.4

Cancel the common factor of

2

$2$ .

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Step 1.4.1

Cancel the common factor.

$\sqrt{6^{\frac{2}{2}} + {(- \sqrt{2})}^{2}}$

Step 1.4.2

Rewrite the expression.

$\sqrt{6^{1} + {(- \sqrt{2})}^{2}}$

$\sqrt{6^{1} + {(- \sqrt{2})}^{2}}$

Step 1.5

Evaluate the exponent.

$\sqrt{6 + {(- \sqrt{2})}^{2}}$

$\sqrt{6 + {(- \sqrt{2})}^{2}}$

Step 2

Simplify the expression.

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Step 2.1

Apply the product rule to

$- \sqrt{2}$ .

$\sqrt{6 + {(- 1)}^{2} {\sqrt{2}}^{2}}$

Step 2.2

Raise

$- 1$ to the power of

$2$ .

$\sqrt{6 + 1 {\sqrt{2}}^{2}}$

Step 2.3

Multiply

${\sqrt{2}}^{2}$ by

$1$ .

$\sqrt{6 + {\sqrt{2}}^{2}}$

$\sqrt{6 + {\sqrt{2}}^{2}}$

Step 3

Rewrite

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

$2$ .

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Step 3.1

Use

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

$\sqrt{2}$ as

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

$\sqrt{6 + {(2^{\frac{1}{2}})}^{2}}$

Step 3.2

Apply the power rule and multiply exponents,

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

$\sqrt{6 + 2^{\frac{1}{2} \cdot 2}}$

Step 3.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{6 + 2^{\frac{2}{2}}}$

Step 3.4

Cancel the common factor of

$2$ .

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Step 3.4.1

Cancel the common factor.

$\sqrt{6 + 2^{\frac{2}{2}}}$

Step 3.4.2

Rewrite the expression.

$\sqrt{6 + 2^{1}}$

$\sqrt{6 + 2^{1}}$

Step 3.5

Evaluate the exponent.

$\sqrt{6 + 2}$

$\sqrt{6 + 2}$

Step 4

Add

$6$ and

$2$ .

$\sqrt{8}$

Step 5

Rewrite

$8$ as

$2^{2} \cdot 2$ .

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Step 5.1

Factor

$4$ out of

$8$ .

$\sqrt{4 (2)}$

Step 5.2

Rewrite

$4$ as

$2^{2}$ .

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

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

Step 6

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$

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