Solve for v 4 = square root of (3v)/(pi*8)

$4 = \sqrt{\frac{3 v}{π \cdot 8}}$

Step 1

Rewrite the equation as $\sqrt{\frac{3 v}{π \cdot 8}} = 4$ .

$\sqrt{\frac{3 v}{π \cdot 8}} = 4$

Step 2

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

${\sqrt{\frac{3 v}{π \cdot 8}}}^{2} = 4^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{3 v}{π \cdot 8}}$ as ${(\frac{3 v}{π \cdot 8})}^{\frac{1}{2}}$ .

${({(\frac{3 v}{π \cdot 8})}^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the left side.

Simplify ${({(\frac{3 v}{π \cdot 8})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{3 v}{π \cdot 8})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{3 v}{π \cdot 8})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{3 v}{π \cdot 8})}^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

${(\frac{3 v}{π \cdot 8})}^{1} = 4^{2}$

Move $8$ to the left of $π$ .

$\frac{3 v}{8 π} = 4^{2}$

Simplify the right side.

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

$\frac{3 v}{8 π} = 16$

Step 4

Solve for $v$ .

Multiply both sides of the equation by $\frac{8 π}{3}$ .

$\frac{8 π}{3} \cdot \frac{3 v}{8 π} = \frac{8 π}{3} \cdot 16$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{8 π}{3} \cdot \frac{3 v}{8 π}$ .

Cancel the common factor of $8 π$ .

Cancel the common factor.

$\frac{8 π}{3} \cdot \frac{3 v}{8 π} = \frac{8 π}{3} \cdot 16$

Rewrite the expression.

$\frac{1}{3} (3 v) = \frac{8 π}{3} \cdot 16$

Cancel the common factor of $3$ .

Factor $3$ out of $3 v$ .

$\frac{1}{3} (3 (v)) = \frac{8 π}{3} \cdot 16$

Cancel the common factor.

$\frac{1}{3} (3 v) = \frac{8 π}{3} \cdot 16$

Rewrite the expression.

$v = \frac{8 π}{3} \cdot 16$

Simplify the right side.

Multiply $\frac{8 π}{3} \cdot 16$ .

Combine $\frac{8 π}{3}$ and $16$ .

$v = \frac{8 π \cdot 16}{3}$

Multiply $16$ by $8$ .

$v = \frac{128 π}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$v = \frac{128 π}{3}$

Decimal Form:

$v = 134.04128655 \dots$

Solve for v 4 = square root of (3v)/(pi*8)

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