Solve for v 4 = square root of v/2

$4 = \sqrt{\frac{v}{2}}$

Rewrite the equation as $\sqrt{\frac{v}{2}} = 4$ .

$\sqrt{\frac{v}{2}} = 4$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{v}{2}}}^{2} = 4^{2}$

Simplify each side of the equation.

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

${({(\frac{v}{2})}^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the left side.

Simplify ${({(\frac{v}{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{v}{2})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{v}{2})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{v}{2})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

${(\frac{v}{2})}^{1} = 4^{2}$

Simplify.

$\frac{v}{2} = 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$\frac{v}{2} = 16$

Solve for $v$ .

Multiply both sides of the equation by $2$ .

$2 \frac{v}{2} = 2 \cdot 16$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{v}{2} = 2 \cdot 16$

Rewrite the expression.

$v = 2 \cdot 16$

Simplify the right side.

Multiply $2$ by $16$ .

$v = 32$