Trigonometry Examples

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

Step 1

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

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $2$ .

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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.

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

Step 1.5

Evaluate the exponent.

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

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 $1$ .

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

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.

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

Step 3.5

Evaluate the exponent.

$\sqrt{6 + 2}$

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 $2^{2}$ .

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

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:

$2.82842712 \dots$