Evaluate square root of (1+cos(80 degrees ))/2

$\sqrt{\frac{1 + \cos (80 °)}{2}}$

Rewrite $\sqrt{\frac{1 + \cos (80 °)}{2}}$ as $\frac{\sqrt{1 + \cos (80 °)}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (80 °)}}{\sqrt{2}}$

Multiply $\frac{\sqrt{1 + \cos (80 °)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (80 °)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{1 + \cos (80 °)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt{1 + \cos (80 °)} \sqrt{2}}{2}$

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt{(1 + \cos (80 °)) \cdot 2}}{2}$

Evaluate $\cos (80 °)$ .

$\frac{\sqrt{(1 + 0.17364817) \cdot 2}}{2}$

Add $1$ and $0.17364817$ .

$\frac{\sqrt{1.17364817 \cdot 2}}{2}$

Multiply $1.17364817$ by $2$ .

$\frac{\sqrt{2.34729635}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt{\frac{1 + \cos (80 °)}{2}}$

Decimal Form:

$0.76604444 \dots$