Calculus Examples

2 p r - \frac{3456}{r^{2}} = 0

$2 p r - \frac{3456}{r^{2}} = 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

1, r^{2}, 1

$1, r^{2}, 1$

Step 1.2

The LCM of one and any expression is the expression.

r^{2}

$r^{2}$

r^{2}

$r^{2}$

Step 2

Multiply each term in

2 p r - \frac{3456}{r^{2}} = 0

$2 p r - \frac{3456}{r^{2}} = 0$ by

r^{2}

$r^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in

2 p r - \frac{3456}{r^{2}} = 0

$2 p r - \frac{3456}{r^{2}} = 0$ by

r^{2}

$r^{2}$ .

2 p r \cdot r^{2} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r \cdot r^{2} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply

r

$r$ by

r^{2}

$r^{2}$ by adding the exponents.

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Step 2.2.1.1.1

Move

r^{2}

$r^{2}$ .

2 p (r^{2} r) - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p (r^{2} r) - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.1.2

Multiply

r^{2}

$r^{2}$ by

r

$r$ .

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Step 2.2.1.1.2.1

Raise

r

$r$ to the power of

1

$1$ .

2 p (r^{2} r^{1}) - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p (r^{2} r^{1}) - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.1.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

2 p r^{2 + 1} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r^{2 + 1} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

2 p r^{2 + 1} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r^{2 + 1} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.1.3

Add

2

$2$ and

1

$1$ .

2 p r^{3} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r^{3} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

2 p r^{3} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r^{3} - \frac{3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.2

Cancel the common factor of

r^{2}

$r^{2}$ .

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Step 2.2.1.2.1

Move the leading negative in

- \frac{3456}{r^{2}}

$- \frac{3456}{r^{2}}$ into the numerator.

2 p r^{3} + \frac{- 3456}{r^{2}} r^{2} = 0 r^{2}

$2 p r^{3} + \frac{- 3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.2.2

Cancel the common factor.

$2 p r^{3} + \frac{- 3456}{r^{2}} r^{2} = 0 r^{2}$

Step 2.2.1.2.3

Rewrite the expression.

$2 p r^{3} - 3456 = 0 r^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply

$0$ by

$r^{2}$ .

$2 p r^{3} - 3456 = 0$

Step 3

Solve the equation.

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Step 3.1

Add

$3456$ to both sides of the equation.

$2 p r^{3} = 3456$

Step 3.2

Divide each term in

$2 p r^{3} = 3456$ by

$2 p$ and simplify.

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Step 3.2.1

Divide each term in

$2 p r^{3} = 3456$ by

$2 p$ .

$\frac{2 p r^{3}}{2 p} = \frac{3456}{2 p}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

$2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 p r^{3}}{2 p} = \frac{3456}{2 p}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{p r^{3}}{p} = \frac{3456}{2 p}$

Step 3.2.2.2

Cancel the common factor of

$p$ .

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Step 3.2.2.2.1

Cancel the common factor.

$\frac{p r^{3}}{p} = \frac{3456}{2 p}$

Step 3.2.2.2.2

Divide

$r^{3}$ by

$1$ .

$r^{3} = \frac{3456}{2 p}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of

$3456$ and

$2$ .

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Step 3.2.3.1.1

Factor

$2$ out of

$3456$ .

$r^{3} = \frac{2 \cdot 1728}{2 p}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor

$2$ out of

$2 p$ .

$r^{3} = \frac{2 \cdot 1728}{2 (p)}$

Step 3.2.3.1.2.2

Cancel the common factor.

$r^{3} = \frac{2 \cdot 1728}{2 p}$

Step 3.2.3.1.2.3

Rewrite the expression.

$r^{3} = \frac{1728}{p}$

Step 3.3

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

$r = \sqrt[3]{\frac{1728}{p}}$

Step 3.4

Simplify

$\sqrt[3]{\frac{1728}{p}}$ .

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Step 3.4.1

Rewrite

$\sqrt[3]{\frac{1728}{p}}$ as

$\frac{\sqrt[3]{1728}}{\sqrt[3]{p}}$ .

$r = \frac{\sqrt[3]{1728}}{\sqrt[3]{p}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Rewrite

$1728$ as

$12^{3}$ .

$r = \frac{\sqrt[3]{12^{3}}}{\sqrt[3]{p}}$

Step 3.4.2.2

Pull terms out from under the radical, assuming real numbers.

$r = \frac{12}{\sqrt[3]{p}}$

Step 3.4.3

Multiply

$\frac{12}{\sqrt[3]{p}}$ by

$\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$r = \frac{12}{\sqrt[3]{p}} \cdot \frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$

Step 3.4.4

Combine and simplify the denominator.

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Step 3.4.4.1

Multiply

$\frac{12}{\sqrt[3]{p}}$ by

$\frac{{\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{2}}$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{\sqrt[3]{p} {\sqrt[3]{p}}^{2}}$

Step 3.4.4.2

Raise

$\sqrt[3]{p}$ to the power of

$1$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1} {\sqrt[3]{p}}^{2}}$

Step 3.4.4.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{12 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{1 + 2}}$

Step 3.4.4.4

Add

$1$ and

$2$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{{\sqrt[3]{p}}^{3}}$

Step 3.4.4.5

Rewrite

${\sqrt[3]{p}}^{3}$ as

$p$ .

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Step 3.4.4.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{p}$ as

$p^{\frac{1}{3}}$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{{(p^{\frac{1}{3}})}^{3}}$

Step 3.4.4.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{p^{\frac{1}{3} \cdot 3}}$

Step 3.4.4.5.3

Combine

$\frac{1}{3}$ and

$3$ .

$r = \frac{12 {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 3.4.4.5.4

Cancel the common factor of

$3$ .

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Step 3.4.4.5.4.1

Cancel the common factor.

$r = \frac{12 {\sqrt[3]{p}}^{2}}{p^{\frac{3}{3}}}$

Step 3.4.4.5.4.2

Rewrite the expression.

$r = \frac{12 {\sqrt[3]{p}}^{2}}{p^{1}}$

Step 3.4.4.5.5

Simplify.

$r = \frac{12 {\sqrt[3]{p}}^{2}}{p}$

Step 3.4.5

Rewrite

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

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

$r = \frac{12 \sqrt[3]{p^{2}}}{p}$