Evaluate square root of (- square root of 3)^2+(-1)^2

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

Step 1

Simplify the expression.

Apply the product rule to $- \sqrt{3}$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply ${\sqrt{3}}^{2}$ by $1$ .

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

Step 2

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$\sqrt{3 + 1}$

Add $3$ and $1$ .

$\sqrt{4}$

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

$\sqrt{2^{2}}$

Step 4

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

$2$