Precalculus Examples

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

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.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

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

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

Step 3.2.1.4.3

Apply the product rule to $2 p$ .

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

Step 3.2.1.5

Simplify the numerator.

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Step 3.2.1.5.1

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

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

Step 3.2.1.5.2

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

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Step 3.2.1.5.2.1

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

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

Step 3.2.1.5.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.2.2.1

Cancel the common factor.

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

Step 3.2.1.5.2.2.2

Rewrite the expression.

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

Step 3.2.1.5.3

Simplify.

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

Step 3.2.1.6

Simplify the denominator.

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Step 3.2.1.6.1

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

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Step 3.2.1.6.1.1

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

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

Step 3.2.1.6.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.6.1.2.1

Cancel the common factor.

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

Step 3.2.1.6.1.2.2

Rewrite the expression.

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

Step 3.2.1.6.2

Simplify.

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

Step 4

Solve for $L$ .

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Step 4.1

Multiply both sides by $g$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $g$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Rewrite the expression.

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

Cancel the common factor.

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

Step 4.3.2.2.2

Divide $L$ by $1$ .

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