Solve for M v = square root of (3RT)/M

$v = \sqrt{\frac{3 R T}{M}}$

Rewrite the equation as $\sqrt{\frac{3 R T}{M}} = v$ .

$\sqrt{\frac{3 R T}{M}} = v$

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

${\sqrt{\frac{3 R T}{M}}}^{2} = v^{2}$

Simplify each side of the equation.

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

${({(\frac{3 R T}{M})}^{\frac{1}{2}})}^{2} = v^{2}$

Simplify the left side.

Simplify ${({(\frac{3 R T}{M})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{3 R T}{M})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{3 R T}{M})}^{\frac{1}{2} \cdot 2} = v^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{3 R T}{M})}^{\frac{1}{2} \cdot 2} = v^{2}$

Rewrite the expression.

${(\frac{3 R T}{M})}^{1} = v^{2}$

Simplify.

$\frac{3 R T}{M} = v^{2}$

Solve for $M$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$M, 1$

The LCM of one and any expression is the expression.

$M$

Multiply each term in $\frac{3 R T}{M} = v^{2}$ by $M$ to eliminate the fractions.

Multiply each term in $\frac{3 R T}{M} = v^{2}$ by $M$ .

$\frac{3 R T}{M} M = v^{2} M$

Simplify the left side.

Cancel the common factor of $M$ .

Cancel the common factor.

$\frac{3 R T}{M} M = v^{2} M$

Rewrite the expression.

$3 R T = v^{2} M$

Solve the equation.

Rewrite the equation as $v^{2} M = 3 R T$ .

$v^{2} M = 3 R T$

Divide each term in $v^{2} M = 3 R T$ by $v^{2}$ and simplify.

Divide each term in $v^{2} M = 3 R T$ by $v^{2}$ .

$\frac{v^{2} M}{v^{2}} = \frac{3 R T}{v^{2}}$

Simplify the left side.

Cancel the common factor of $v^{2}$ .

Cancel the common factor.

$\frac{v^{2} M}{v^{2}} = \frac{3 R T}{v^{2}}$

Divide $M$ by $1$ .

$M = \frac{3 R T}{v^{2}}$

Solve for M v = square root of (3RT)/M

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