Trigonometry Examples

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

Step 1

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

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$\sqrt{- 3^{1} + 1^{2}}$

Step 1.5

Evaluate the exponent.

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

Step 2

Simplify the expression.

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Step 2.1

Multiply $- 1$ by $3$ .

$\sqrt{- 3 + 1^{2}}$

Step 2.2

One to any power is one.

$\sqrt{- 3 + 1}$

Step 2.3

Add $- 3$ and $1$ .

$\sqrt{- 2}$

Step 3

Rewrite $- 2$ as $- 1 (2)$ .

$\sqrt{- 1 (2)}$

Step 4

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$\sqrt{- 1} \cdot \sqrt{2}$

Step 5

Rewrite $\sqrt{- 1}$ as $i$ .

$i \sqrt{2}$