Simplify square root of (1+cos(7A))/2

$\sqrt{\frac{1 + \cos (7 A)}{2}}$

Step 1

Rewrite $\sqrt{\frac{1 + \cos (7 A)}{2}}$ as $\frac{\sqrt{1 + \cos (7 A)}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (7 A)}}{\sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{1 + \cos (7 A)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (7 A)}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{\sqrt{1 + \cos (7 A)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt{1 + \cos (7 A)} \sqrt{2}}{2}$

Step 4

Combine using the product rule for radicals.

$\frac{\sqrt{(1 + \cos (7 A)) \cdot 2}}{2}$

Step 5

Reorder factors in $\frac{\sqrt{(1 + \cos (7 A)) \cdot 2}}{2}$ .

$\frac{\sqrt{2 (1 + \cos (7 A))}}{2}$