Solve for t s=1/3*(gt^2)

$s = \frac{1}{3} \cdot (g t^{2})$

Step 1

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

$\frac{1}{3} \cdot (g t^{2}) = s$

Step 2

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot (g t^{2})) = 3 s$

Step 3

Simplify the left side.

Simplify $3 (\frac{1}{3} \cdot (g t^{2}))$ .

Multiply $\frac{1}{3} (g t^{2})$ .

Combine $g$ and $\frac{1}{3}$ .

$3 (\frac{g}{3} t^{2}) = 3 s$

Combine $\frac{g}{3}$ and $t^{2}$ .

$3 \frac{g t^{2}}{3} = 3 s$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{g t^{2}}{3} = 3 s$

Rewrite the expression.

$g t^{2} = 3 s$

Step 4

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

Divide each term in $g t^{2} = 3 s$ by $g$ .

$\frac{g t^{2}}{g} = \frac{3 s}{g}$

Simplify the left side.

Cancel the common factor of $g$ .

Cancel the common factor.

$\frac{g t^{2}}{g} = \frac{3 s}{g}$

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

$t^{2} = \frac{3 s}{g}$

Step 5

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

$t = \pm \sqrt{\frac{3 s}{g}}$

Step 6

Simplify $\pm \sqrt{\frac{3 s}{g}}$ .

Rewrite $\sqrt{\frac{3 s}{g}}$ as $\frac{\sqrt{3 s}}{\sqrt{g}}$ .

$t = \pm \frac{\sqrt{3 s}}{\sqrt{g}}$

Multiply $\frac{\sqrt{3 s}}{\sqrt{g}}$ by $\frac{\sqrt{g}}{\sqrt{g}}$ .

$t = \pm \frac{\sqrt{3 s}}{\sqrt{g}} \cdot \frac{\sqrt{g}}{\sqrt{g}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 s}}{\sqrt{g}}$ by $\frac{\sqrt{g}}{\sqrt{g}}$ .

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{\sqrt{g} \sqrt{g}}$

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{{\sqrt{g}}^{1} \sqrt{g}}$

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{{\sqrt{g}}^{1} {\sqrt{g}}^{1}}$

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{{\sqrt{g}}^{1 + 1}}$

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{{\sqrt{g}}^{2}}$

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

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{{(g^{\frac{1}{2}})}^{2}}$

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{g^{\frac{1}{2} \cdot 2}}$

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

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{g^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{g^{\frac{2}{2}}}$

Rewrite the expression.

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{g^{1}}$

Simplify.

$t = \pm \frac{\sqrt{3 s} \sqrt{g}}{g}$

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{3 s g}}{g}$

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.

$t = \frac{\sqrt{3 s g}}{g}$

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

$t = - \frac{\sqrt{3 s g}}{g}$

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

$t = \frac{\sqrt{3 s g}}{g}$

$t = - \frac{\sqrt{3 s g}}{g}$

$t = \frac{\sqrt{3 s g}}{g}$

$t = - \frac{\sqrt{3 s g}}{g}$

Solve for t s=1/3*(gt^2)

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