Solve for g t=2p square root of l/g

$t = 2 p \sqrt{\frac{l}{g}}$

Step 1

Rewrite the equation as $2 p \sqrt{\frac{l}{g}} = t$ .

$2 p \sqrt{\frac{l}{g}} = t$

Step 2

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

${(2 p \sqrt{\frac{l}{g}})}^{2} = t^{2}$

Step 3

Simplify each side of the equation.

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

${(2 p {(\frac{l}{g})}^{\frac{1}{2}})}^{2} = t^{2}$

Simplify the left side.

Simplify ${(2 p {(\frac{l}{g})}^{\frac{1}{2}})}^{2}$ .

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

${(2 p \frac{l^{\frac{1}{2}}}{g^{\frac{1}{2}}})}^{2} = t^{2}$

Multiply $2 p \frac{l^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ .

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

${(p \frac{2 l^{\frac{1}{2}}}{g^{\frac{1}{2}}})}^{2} = t^{2}$

Combine $p$ and $\frac{2 l^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ .

${(\frac{p (2 l^{\frac{1}{2}})}{g^{\frac{1}{2}}})}^{2} = t^{2}$

Move $2$ to the left of $p$ .

${(\frac{2 \cdot p l^{\frac{1}{2}}}{g^{\frac{1}{2}}})}^{2} = t^{2}$

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

Apply the product rule to $\frac{2 p l^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ .

$\frac{{(2 p l^{\frac{1}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Apply the product rule to $2 p l^{\frac{1}{2}}$ .

$\frac{{(2 p)}^{2} {(l^{\frac{1}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Apply the product rule to $2 p$ .

$\frac{2^{2} p^{2} {(l^{\frac{1}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Simplify the numerator.

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

$\frac{4 p^{2} {(l^{\frac{1}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

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

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

$\frac{4 p^{2} l^{\frac{1}{2} \cdot 2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 p^{2} l^{\frac{1}{2} \cdot 2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Rewrite the expression.

$\frac{4 p^{2} l^{1}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Simplify.

$\frac{4 p^{2} l}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Simplify the denominator.

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

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

$\frac{4 p^{2} l}{g^{\frac{1}{2} \cdot 2}} = t^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 p^{2} l}{g^{\frac{1}{2} \cdot 2}} = t^{2}$

Rewrite the expression.

$\frac{4 p^{2} l}{g^{1}} = t^{2}$

Simplify.

$\frac{4 p^{2} l}{g} = t^{2}$

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 $\frac{4 p^{2} l}{g} = t^{2}$ by $g$ to eliminate the fractions.

Multiply each term in $\frac{4 p^{2} l}{g} = t^{2}$ by $g$ .

$\frac{4 p^{2} l}{g} g = t^{2} g$

Simplify the left side.

Cancel the common factor of $g$ .

Cancel the common factor.

$\frac{4 p^{2} l}{g} g = t^{2} g$

Rewrite the expression.

$4 p^{2} l = t^{2} g$

Solve the equation.

Rewrite the equation as $t^{2} g = 4 p^{2} l$ .

$t^{2} g = 4 p^{2} l$

Divide each term in $t^{2} g = 4 p^{2} l$ by $t^{2}$ and simplify.

Divide each term in $t^{2} g = 4 p^{2} l$ by $t^{2}$ .

$\frac{t^{2} g}{t^{2}} = \frac{4 p^{2} l}{t^{2}}$

Simplify the left side.

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

Cancel the common factor.

$\frac{t^{2} g}{t^{2}} = \frac{4 p^{2} l}{t^{2}}$

Divide $g$ by $1$ .

$g = \frac{4 p^{2} l}{t^{2}}$

Solve for g t=2p square root of l/g

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