Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

$\sqrt{4 {\sqrt{2}}^{2} + {(- 2 \sqrt{2})}^{2}}$

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

$\sqrt{4 {(2^{\frac{1}{2}})}^{2} + {(- 2 \sqrt{2})}^{2}}$

Step 2.2

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

$\sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} + {(- 2 \sqrt{2})}^{2}}$

Step 2.3

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

$\sqrt{4 \cdot 2^{\frac{2}{2}} + {(- 2 \sqrt{2})}^{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$\sqrt{4 \cdot 2^{\frac{2}{2}} + {(- 2 \sqrt{2})}^{2}}$

Step 2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 2^{1} + {(- 2 \sqrt{2})}^{2}}$

Step 2.5

Evaluate the exponent.

$\sqrt{4 \cdot 2 + {(- 2 \sqrt{2})}^{2}}$

Step 3

Simplify the expression.

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Step 3.1

Multiply $4$ by $2$ .

$\sqrt{8 + {(- 2 \sqrt{2})}^{2}}$

Step 3.2

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

$\sqrt{8 + {(- 2)}^{2} {\sqrt{2}}^{2}}$

Step 3.3

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

$\sqrt{8 + 4 {\sqrt{2}}^{2}}$

Step 4

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

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Step 4.1

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

$\sqrt{8 + 4 {(2^{\frac{1}{2}})}^{2}}$

Step 4.2

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

$\sqrt{8 + 4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.3

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

$\sqrt{8 + 4 \cdot 2^{\frac{2}{2}}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\sqrt{8 + 4 \cdot 2^{\frac{2}{2}}}$

Step 4.4.2

Rewrite the expression.

$\sqrt{8 + 4 \cdot 2^{1}}$

Step 4.5

Evaluate the exponent.

$\sqrt{8 + 4 \cdot 2}$

Step 5

Simplify the expression.

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Step 5.1

Multiply $4$ by $2$ .

$\sqrt{8 + 8}$

Step 5.2

Add $8$ and $8$ .

$\sqrt{16}$

Step 5.3

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

$\sqrt{4^{2}}$

Step 5.4

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

$4$