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

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

Step 1

Simplify the expression.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

$\sqrt{\frac{1}{4} + {(\frac{\sqrt{3}}{2})}^{2}}$

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2

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

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

$\sqrt{\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

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

$\sqrt{\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

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

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Rewrite the expression.

$\sqrt{\frac{1}{4} + \frac{3^{1}}{2^{2}}}$

Evaluate the exponent.

$\sqrt{\frac{1}{4} + \frac{3}{2^{2}}}$

Step 3

Simplify the expression.

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

$\sqrt{\frac{1}{4} + \frac{3}{4}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 3}{4}}$

Add $1$ and $3$ .

$\sqrt{\frac{4}{4}}$

Divide $4$ by $4$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$