Solve for g T=9pi square root of L/g

$T = 9 π \sqrt{\frac{L}{g}}$

Step 1

Rewrite the equation as $9 π \sqrt{\frac{L}{g}} = T$ .

$9 π \sqrt{\frac{L}{g}} = T$

Step 2

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

${(9 π \sqrt{\frac{L}{g}})}^{2} = T^{2}$

Step 3

Simplify each side of the equation.

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

${(9 π {(\frac{L}{g})}^{\frac{1}{2}})}^{2} = T^{2}$

Simplify the left side.

Simplify ${(9 π {(\frac{L}{g})}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $\frac{L}{g}$ .

${(9 π \frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Multiply $9 π \frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ .

Combine $\frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ and $9$ .

${(\frac{L^{\frac{1}{2}} \cdot 9}{g^{\frac{1}{2}}} π)}^{2} = T^{2}$

Combine $\frac{L^{\frac{1}{2}} \cdot 9}{g^{\frac{1}{2}}}$ and $π$ .

${(\frac{L^{\frac{1}{2}} \cdot 9 π}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Move $9$ to the left of $L^{\frac{1}{2}}$ .

${(\frac{9 \cdot L^{\frac{1}{2}} π}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{9 L^{\frac{1}{2}} π}{g^{\frac{1}{2}}}$ .

$\frac{{(9 L^{\frac{1}{2}} π)}^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Apply the product rule to $9 L^{\frac{1}{2}} π$ .

$\frac{{(9 L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Apply the product rule to $9 L^{\frac{1}{2}}$ .

$\frac{9^{2} {(L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Simplify the numerator.

Raise $9$ to the power of $2$ .

$\frac{81 {(L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

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

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

$\frac{81 L^{\frac{1}{2} \cdot 2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{81 L^{\frac{1}{2} \cdot 2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Rewrite the expression.

$\frac{81 L^{1} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Simplify.

$\frac{81 L π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Simplify the denominator.

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

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

$\frac{81 L π^{2}}{g^{\frac{1}{2} \cdot 2}} = T^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{81 L π^{2}}{g^{\frac{1}{2} \cdot 2}} = T^{2}$

Rewrite the expression.

$\frac{81 L π^{2}}{g^{1}} = T^{2}$

Simplify.

$\frac{81 L π^{2}}{g} = T^{2}$

Step 4

Solve for $g$ .

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.

$g, 1$

The LCM of one and any expression is the expression.

$g$

Multiply each term in $\frac{81 L π^{2}}{g} = T^{2}$ by $g$ to eliminate the fractions.

Multiply each term in $\frac{81 L π^{2}}{g} = T^{2}$ by $g$ .

$\frac{81 L π^{2}}{g} g = T^{2} g$

Simplify the left side.

Cancel the common factor of $g$ .

Cancel the common factor.

$\frac{81 L π^{2}}{g} g = T^{2} g$

Rewrite the expression.

$81 L π^{2} = T^{2} g$

Solve the equation.

Rewrite the equation as $T^{2} g = 81 L π^{2}$ .

$T^{2} g = 81 L π^{2}$

Divide each term in $T^{2} g = 81 L π^{2}$ by $T^{2}$ and simplify.

Divide each term in $T^{2} g = 81 L π^{2}$ by $T^{2}$ .

$\frac{T^{2} g}{T^{2}} = \frac{81 L π^{2}}{T^{2}}$

Simplify the left side.

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

Cancel the common factor.

$\frac{T^{2} g}{T^{2}} = \frac{81 L π^{2}}{T^{2}}$

Divide $g$ by $1$ .

$g = \frac{81 L π^{2}}{T^{2}}$

Solve for g T=9pi square root of L/g

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