Basic Math Examples

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

Step 1

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

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

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

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

Step 3.2.1.2.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

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

Step 3.2.1.4.2

Raise $l$ to the power of $1$ .

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

Step 3.2.1.4.3

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

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

Step 3.2.1.4.4

Write $1$ as a fraction with a common denominator.

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

Step 3.2.1.4.5

Combine the numerators over the common denominator.

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

Step 3.2.1.4.6

Add $2$ and $1$ .

${(\frac{2 p l^{\frac{3}{2}}}{g^{\frac{1}{2}}})}^{2} = t^{2}$

Step 3.2.1.5

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

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Step 3.2.1.5.1

Apply the product rule to $\frac{2 p l^{\frac{3}{2}}}{g^{\frac{1}{2}}}$ .

$\frac{{(2 p l^{\frac{3}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.5.2

Apply the product rule to $2 p l^{\frac{3}{2}}$ .

$\frac{{(2 p)}^{2} {(l^{\frac{3}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.5.3

Apply the product rule to $2 p$ .

$\frac{2^{2} p^{2} {(l^{\frac{3}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.6

Simplify the numerator.

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Step 3.2.1.6.1

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

$\frac{4 p^{2} {(l^{\frac{3}{2}})}^{2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.6.2

Multiply the exponents in ${(l^{\frac{3}{2}})}^{2}$ .

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Step 3.2.1.6.2.1

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

$\frac{4 p^{2} l^{\frac{3}{2} \cdot 2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.6.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.6.2.2.1

Cancel the common factor.

$\frac{4 p^{2} l^{\frac{3}{2} \cdot 2}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.6.2.2.2

Rewrite the expression.

$\frac{4 p^{2} l^{3}}{{(g^{\frac{1}{2}})}^{2}} = t^{2}$

Step 3.2.1.7

Simplify the denominator.

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Step 3.2.1.7.1

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

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Step 3.2.1.7.1.1

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

$\frac{4 p^{2} l^{3}}{g^{\frac{1}{2} \cdot 2}} = t^{2}$

Step 3.2.1.7.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.7.1.2.1

Cancel the common factor.

$\frac{4 p^{2} l^{3}}{g^{\frac{1}{2} \cdot 2}} = t^{2}$

Step 3.2.1.7.1.2.2

Rewrite the expression.

$\frac{4 p^{2} l^{3}}{g^{1}} = t^{2}$

Step 3.2.1.7.2

Simplify.

$\frac{4 p^{2} l^{3}}{g} = t^{2}$

Step 4

Solve for $l$ .

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Step 4.1

Multiply both sides by $g$ .

$\frac{4 p^{2} l^{3}}{g} g = t^{2} g$

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.

$\frac{4 p^{2} l^{3}}{g} g = t^{2} g$

Step 4.2.1.2

Rewrite the expression.

$4 p^{2} l^{3} = t^{2} g$

Step 4.3

Solve for $l$ .

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Step 4.3.1

Divide each term in $4 p^{2} l^{3} = t^{2} g$ by $4 p^{2}$ and simplify.

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Step 4.3.1.1

Divide each term in $4 p^{2} l^{3} = t^{2} g$ by $4 p^{2}$ .

$\frac{4 p^{2} l^{3}}{4 p^{2}} = \frac{t^{2} g}{4 p^{2}}$

Step 4.3.1.2

Simplify the left side.

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Step 4.3.1.2.1

Cancel the common factor of $4$ .

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Step 4.3.1.2.1.1

Cancel the common factor.

$\frac{4 p^{2} l^{3}}{4 p^{2}} = \frac{t^{2} g}{4 p^{2}}$

Step 4.3.1.2.1.2

Rewrite the expression.

$\frac{p^{2} l^{3}}{p^{2}} = \frac{t^{2} g}{4 p^{2}}$

Step 4.3.1.2.2

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

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Step 4.3.1.2.2.1

Cancel the common factor.

$\frac{p^{2} l^{3}}{p^{2}} = \frac{t^{2} g}{4 p^{2}}$

Step 4.3.1.2.2.2

Divide $l^{3}$ by $1$ .

$l^{3} = \frac{t^{2} g}{4 p^{2}}$

Step 4.3.2

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

$l = \sqrt[3]{\frac{t^{2} g}{4 p^{2}}}$

Step 4.3.3

Simplify $\sqrt[3]{\frac{t^{2} g}{4 p^{2}}}$ .

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Step 4.3.3.1

Rewrite $\sqrt[3]{\frac{t^{2} g}{4 p^{2}}}$ as $\frac{\sqrt[3]{t^{2} g}}{\sqrt[3]{4 p^{2}}}$ .

$l = \frac{\sqrt[3]{t^{2} g}}{\sqrt[3]{4 p^{2}}}$

Step 4.3.3.2

Multiply $\frac{\sqrt[3]{t^{2} g}}{\sqrt[3]{4 p^{2}}}$ by $\frac{{\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{2}}$ .

$l = \frac{\sqrt[3]{t^{2} g}}{\sqrt[3]{4 p^{2}}} \cdot \frac{{\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{2}}$

Step 4.3.3.3

Combine and simplify the denominator.

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Step 4.3.3.3.1

Multiply $\frac{\sqrt[3]{t^{2} g}}{\sqrt[3]{4 p^{2}}}$ by $\frac{{\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{2}}$ .

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{\sqrt[3]{4 p^{2}} {\sqrt[3]{4 p^{2}}}^{2}}$

Step 4.3.3.3.2

Raise $\sqrt[3]{4 p^{2}}$ to the power of $1$ .

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{1} {\sqrt[3]{4 p^{2}}}^{2}}$

Step 4.3.3.3.3

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

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{1 + 2}}$

Step 4.3.3.3.4

Add $1$ and $2$ .

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{\sqrt[3]{4 p^{2}}}^{3}}$

Step 4.3.3.3.5

Rewrite ${\sqrt[3]{4 p^{2}}}^{3}$ as $4 p^{2}$ .

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Step 4.3.3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 p^{2}}$ as ${(4 p^{2})}^{\frac{1}{3}}$ .

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{({(4 p^{2})}^{\frac{1}{3}})}^{3}}$

Step 4.3.3.3.5.2

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

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{(4 p^{2})}^{\frac{1}{3} \cdot 3}}$

Step 4.3.3.3.5.3

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

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{(4 p^{2})}^{\frac{3}{3}}}$

Step 4.3.3.3.5.4

Cancel the common factor of $3$ .

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Step 4.3.3.3.5.4.1

Cancel the common factor.

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{(4 p^{2})}^{\frac{3}{3}}}$

Step 4.3.3.3.5.4.2

Rewrite the expression.

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{{(4 p^{2})}^{1}}$

Step 4.3.3.3.5.5

Simplify.

$l = \frac{\sqrt[3]{t^{2} g} {\sqrt[3]{4 p^{2}}}^{2}}{4 p^{2}}$

Step 4.3.3.4

Simplify the numerator.

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Step 4.3.3.4.1

Rewrite ${\sqrt[3]{4 p^{2}}}^{2}$ as $\sqrt[3]{{(4 p^{2})}^{2}}$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{{(4 p^{2})}^{2}}}{4 p^{2}}$

Step 4.3.3.4.2

Apply the product rule to $4 p^{2}$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{4^{2} {(p^{2})}^{2}}}{4 p^{2}}$

Step 4.3.3.4.3

Raise $4$ to the power of $2$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{16 {(p^{2})}^{2}}}{4 p^{2}}$

Step 4.3.3.4.4

Multiply the exponents in ${(p^{2})}^{2}$ .

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Step 4.3.3.4.4.1

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

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{16 p^{2 \cdot 2}}}{4 p^{2}}$

Step 4.3.3.4.4.2

Multiply $2$ by $2$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{16 p^{4}}}{4 p^{2}}$

Step 4.3.3.4.5

Rewrite $16 p^{4}$ as ${(2 p)}^{3} \cdot (2 p)$ .

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Step 4.3.3.4.5.1

Factor $8$ out of $16$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{8 (2) p^{4}}}{4 p^{2}}$

Step 4.3.3.4.5.2

Rewrite $8$ as $2^{3}$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{2^{3} \cdot 2 p^{4}}}{4 p^{2}}$

Step 4.3.3.4.5.3

Factor out $p^{3}$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{2^{3} \cdot 2 (p^{3} p)}}{4 p^{2}}$

Step 4.3.3.4.5.4

Move $2$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{2^{3} p^{3} \cdot 2 p}}{4 p^{2}}$

Step 4.3.3.4.5.5

Rewrite $2^{3} p^{3}$ as ${(2 p)}^{3}$ .

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{{(2 p)}^{3} \cdot 2 p}}{4 p^{2}}$

Step 4.3.3.4.5.6

Add parentheses.

$l = \frac{\sqrt[3]{t^{2} g} \sqrt[3]{{(2 p)}^{3} \cdot (2 p)}}{4 p^{2}}$

Step 4.3.3.4.6

Pull terms out from under the radical.

$l = \frac{\sqrt[3]{t^{2} g} \cdot 2 p \sqrt[3]{2 p}}{4 p^{2}}$

Step 4.3.3.4.7

Combine using the product rule for radicals.

$l = \frac{2 p \sqrt[3]{t^{2} g \cdot 2 p}}{4 p^{2}}$

Step 4.3.3.5

Reduce the expression by cancelling the common factors.

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Step 4.3.3.5.1

Cancel the common factor of $2$ and $4$ .

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Step 4.3.3.5.1.1

Factor $2$ out of $2 p \sqrt[3]{t^{2} g \cdot 2 p}$ .

$l = \frac{2 (p \sqrt[3]{t^{2} g \cdot 2 p})}{4 p^{2}}$

Step 4.3.3.5.1.2

Cancel the common factors.

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Step 4.3.3.5.1.2.1

Factor $2$ out of $4 p^{2}$ .

$l = \frac{2 (p \sqrt[3]{t^{2} g \cdot 2 p})}{2 (2 p^{2})}$

Step 4.3.3.5.1.2.2

Cancel the common factor.

$l = \frac{2 (p \sqrt[3]{t^{2} g \cdot 2 p})}{2 (2 p^{2})}$

Step 4.3.3.5.1.2.3

Rewrite the expression.

$l = \frac{p \sqrt[3]{t^{2} g \cdot 2 p}}{2 p^{2}}$

Step 4.3.3.5.2

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

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Step 4.3.3.5.2.1

Factor $p$ out of $p \sqrt[3]{t^{2} g \cdot 2 p}$ .

$l = \frac{p (\sqrt[3]{t^{2} g \cdot 2 p})}{2 p^{2}}$

Step 4.3.3.5.2.2

Cancel the common factors.

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Step 4.3.3.5.2.2.1

Factor $p$ out of $2 p^{2}$ .

$l = \frac{p (\sqrt[3]{t^{2} g \cdot 2 p})}{p (2 p)}$

Step 4.3.3.5.2.2.2

Cancel the common factor.

$l = \frac{p \sqrt[3]{t^{2} g \cdot 2 p}}{p (2 p)}$

Step 4.3.3.5.2.2.3

Rewrite the expression.

$l = \frac{\sqrt[3]{t^{2} g \cdot 2 p}}{2 p}$

Step 4.3.3.5.3

Reorder factors in $\frac{\sqrt[3]{t^{2} g \cdot 2 p}}{2 p}$ .

$l = \frac{\sqrt[3]{2 t^{2} g p}}{2 p}$