Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Evaluate the exponent.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Evaluate the exponent.

$\sqrt{2 + 2}$

Step 3

Simplify the expression.

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Step 3.1

Add $2$ and $2$ .

$\sqrt{4}$

Step 3.2

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

$\sqrt{2^{2}}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$2$