Solve for s V=1/3s^2h

$V = \frac{1}{3} s^{2} h$

Step 1

Rewrite the equation as $\frac{1}{3} \cdot (s^{2} h) = V$ .

$\frac{1}{3} \cdot (s^{2} h) = V$

Step 2

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot (s^{2} h)) = 3 V$

Step 3

Simplify the left side.

Simplify $3 (\frac{1}{3} \cdot (s^{2} h))$ .

Multiply $\frac{1}{3} (s^{2} h)$ .

Combine $s^{2}$ and $\frac{1}{3}$ .

$3 (\frac{s^{2}}{3} h) = 3 V$

Combine $\frac{s^{2}}{3}$ and $h$ .

$3 \frac{s^{2} h}{3} = 3 V$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{s^{2} h}{3} = 3 V$

Rewrite the expression.

$s^{2} h = 3 V$

Step 4

Divide each term in $s^{2} h = 3 V$ by $h$ and simplify.

Divide each term in $s^{2} h = 3 V$ by $h$ .

$\frac{s^{2} h}{h} = \frac{3 V}{h}$

Simplify the left side.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{s^{2} h}{h} = \frac{3 V}{h}$

Divide $s^{2}$ by $1$ .

$s^{2} = \frac{3 V}{h}$

Step 5

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$s = \pm \sqrt{\frac{3 V}{h}}$

Step 6

Simplify $\pm \sqrt{\frac{3 V}{h}}$ .

Rewrite $\sqrt{\frac{3 V}{h}}$ as $\frac{\sqrt{3 V}}{\sqrt{h}}$ .

$s = \pm \frac{\sqrt{3 V}}{\sqrt{h}}$

Multiply $\frac{\sqrt{3 V}}{\sqrt{h}}$ by $\frac{\sqrt{h}}{\sqrt{h}}$ .

$s = \pm \frac{\sqrt{3 V}}{\sqrt{h}} \cdot \frac{\sqrt{h}}{\sqrt{h}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 V}}{\sqrt{h}}$ by $\frac{\sqrt{h}}{\sqrt{h}}$ .

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{\sqrt{h} \sqrt{h}}$

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{{\sqrt{h}}^{1} \sqrt{h}}$

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{{\sqrt{h}}^{1} {\sqrt{h}}^{1}}$

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{{\sqrt{h}}^{1 + 1}}$

Add $1$ and $1$ .

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{{\sqrt{h}}^{2}}$

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

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{{(h^{\frac{1}{2}})}^{2}}$

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{h^{\frac{1}{2} \cdot 2}}$

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

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{h^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{h^{\frac{2}{2}}}$

Rewrite the expression.

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{h^{1}}$

Simplify.

$s = \pm \frac{\sqrt{3 V} \sqrt{h}}{h}$

Combine using the product rule for radicals.

$s = \pm \frac{\sqrt{3 V h}}{h}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$s = \frac{\sqrt{3 V h}}{h}$

Next, use the negative value of the $\pm$ to find the second solution.

$s = - \frac{\sqrt{3 V h}}{h}$

The complete solution is the result of both the positive and negative portions of the solution.

$s = \frac{\sqrt{3 V h}}{h}$

$s = - \frac{\sqrt{3 V h}}{h}$

$s = \frac{\sqrt{3 V h}}{h}$

$s = - \frac{\sqrt{3 V h}}{h}$

Solve for s V=1/3s^2h

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