Calculus Examples

$5 L = 20480 \cdot \frac{L^{4}}{4096^{2}}$

Step 1

Simplify $20480 \cdot \frac{L^{4}}{4096^{2}}$ .

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Step 1.1

Raise $4096$ to the power of $2$ .

$5 L = 20480 \cdot \frac{L^{4}}{16777216}$

Step 1.2

Cancel the common factor of $4096$ .

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Step 1.2.1

Factor $4096$ out of $20480$ .

$5 L = 4096 (5) \cdot \frac{L^{4}}{16777216}$

Step 1.2.2

Factor $4096$ out of $16777216$ .

$5 L = 4096 \cdot 5 \cdot \frac{L^{4}}{4096 \cdot 4096}$

Step 1.2.3

Cancel the common factor.

$5 L = 4096 \cdot 5 \cdot \frac{L^{4}}{4096 \cdot 4096}$

Step 1.2.4

Rewrite the expression.

$5 L = 5 \cdot \frac{L^{4}}{4096}$

$5 L = \frac{5 L^{4}}{4096}$

Step 2

Subtract $\frac{5 L^{4}}{4096}$ from both sides of the equation.

$5 L - \frac{5 L^{4}}{4096} = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $5 L$ out of $5 L - \frac{5 L^{4}}{4096}$ .

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Step 3.1.1

Factor $5 L$ out of $5 L$ .

$5 L (1) - \frac{5 L^{4}}{4096} = 0$

Step 3.1.2

Factor $5 L$ out of $- \frac{5 L^{4}}{4096}$ .

$5 L (1) + 5 L (- \frac{L^{3}}{4096}) = 0$

Step 3.1.3

Factor $5 L$ out of $5 L (1) + 5 L (- \frac{L^{3}}{4096})$ .

$5 L (1 - \frac{L^{3}}{4096}) = 0$

Step 3.2

Rewrite $1$ as $1^{3}$ .

$5 L (1^{3} - \frac{L^{3}}{4096}) = 0$

Step 3.3

Rewrite $\frac{L^{3}}{4096}$ as ${(\frac{L}{16})}^{3}$ .

$5 L (1^{3} - {(\frac{L}{16})}^{3}) = 0$

Step 3.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = \frac{L}{16}$ .

$5 L ((1 - \frac{L}{16}) (1^{2} + 1 (\frac{L}{16}) + {(\frac{L}{16})}^{2})) = 0$

Step 3.5

Factor.

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Step 3.5.1

Simplify.

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Step 3.5.1.1

One to any power is one.

$5 L ((1 - \frac{L}{16}) (1 + 1 (\frac{L}{16}) + {(\frac{L}{16})}^{2})) = 0$

Step 3.5.1.2

Multiply $\frac{L}{16}$ by $1$ .

$5 L ((1 - \frac{L}{16}) (1 + \frac{L}{16} + {(\frac{L}{16})}^{2})) = 0$

Step 3.5.1.3

Apply the product rule to $\frac{L}{16}$ .

$5 L ((1 - \frac{L}{16}) (1 + \frac{L}{16} + \frac{L^{2}}{16^{2}})) = 0$

Step 3.5.1.4

Raise $16$ to the power of $2$ .

$5 L ((1 - \frac{L}{16}) (1 + \frac{L}{16} + \frac{L^{2}}{256})) = 0$

Step 3.5.2

Remove unnecessary parentheses.

$5 L (1 - \frac{L}{16}) (1 + \frac{L}{16} + \frac{L^{2}}{256}) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$L = 0$

$1 - \frac{L}{16} = 0$

$1 + \frac{L}{16} + \frac{L^{2}}{256} = 0$

Step 5

Set $L$ equal to $0$ .

$L = 0$

Step 6

Set $1 - \frac{L}{16}$ equal to $0$ and solve for $L$ .

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Step 6.1

Set $1 - \frac{L}{16}$ equal to $0$ .

$1 - \frac{L}{16} = 0$

Step 6.2

Solve $1 - \frac{L}{16} = 0$ for $L$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$- \frac{L}{16} = - 1$

Step 6.2.2

Multiply both sides of the equation by $- 16$ .

$- 16 (- \frac{L}{16}) = - 16 \cdot - 1$

Step 6.2.3

Simplify both sides of the equation.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify $- 16 (- \frac{L}{16})$ .

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Step 6.2.3.1.1.1

Cancel the common factor of $16$ .

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Step 6.2.3.1.1.1.1

Move the leading negative in $- \frac{L}{16}$ into the numerator.

$- 16 \frac{- L}{16} = - 16 \cdot - 1$

Step 6.2.3.1.1.1.2

Factor $16$ out of $- 16$ .

$16 (- 1) \frac{- L}{16} = - 16 \cdot - 1$

Step 6.2.3.1.1.1.3

Cancel the common factor.

$16 \cdot - 1 \frac{- L}{16} = - 16 \cdot - 1$

Step 6.2.3.1.1.1.4

Rewrite the expression.

$- - L = - 16 \cdot - 1$

Step 6.2.3.1.1.2

Multiply.

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Step 6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 L = - 16 \cdot - 1$

Step 6.2.3.1.1.2.2

Multiply $L$ by $1$ .

$L = - 16 \cdot - 1$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Multiply $- 16$ by $- 1$ .

$L = 16$

Step 7

Set $1 + \frac{L}{16} + \frac{L^{2}}{256}$ equal to $0$ and solve for $L$ .

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Step 7.1

Set $1 + \frac{L}{16} + \frac{L^{2}}{256}$ equal to $0$ .

$1 + \frac{L}{16} + \frac{L^{2}}{256} = 0$

Step 7.2

Solve $1 + \frac{L}{16} + \frac{L^{2}}{256} = 0$ for $L$ .

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Step 7.2.1

Multiply through by the least common denominator $256$ , then simplify.

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Step 7.2.1.1

Apply the distributive property.

$256 \cdot 1 + 256 (\frac{L}{16}) + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2

Simplify.

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Step 7.2.1.2.1

Multiply $256$ by $1$ .

$256 + 256 (\frac{L}{16}) + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2.2

Cancel the common factor of $16$ .

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Step 7.2.1.2.2.1

Factor $16$ out of $256$ .

$256 + 16 (16) (\frac{L}{16}) + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2.2.2

Cancel the common factor.

$256 + 16 \cdot (16 (\frac{L}{16})) + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2.2.3

Rewrite the expression.

$256 + 16 L + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2.3

Cancel the common factor of $256$ .

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Step 7.2.1.2.3.1

Cancel the common factor.

$256 + 16 L + 256 (\frac{L^{2}}{256}) = 0$

Step 7.2.1.2.3.2

Rewrite the expression.

$256 + 16 L + L^{2} = 0$

Step 7.2.1.3

Move $256$ .

$16 L + L^{2} + 256 = 0$

Step 7.2.1.4

Reorder $16 L$ and $L^{2}$ .

$L^{2} + 16 L + 256 = 0$

Step 7.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.3

Substitute the values $a = 1$ , $b = 16$ , and $c = 256$ into the quadratic formula and solve for $L$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot 256)}}{2 \cdot 1}$

Step 7.2.4

Simplify.

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Step 7.2.4.1

Simplify the numerator.

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Step 7.2.4.1.1

Raise $16$ to the power of $2$ .

$L = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 256}}{2 \cdot 1}$

Step 7.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 256$ .

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Step 7.2.4.1.2.1

Multiply $- 4$ by $1$ .

$L = \frac{- 16 \pm \sqrt{256 - 4 \cdot 256}}{2 \cdot 1}$

Step 7.2.4.1.2.2

Multiply $- 4$ by $256$ .

$L = \frac{- 16 \pm \sqrt{256 - 1024}}{2 \cdot 1}$

Step 7.2.4.1.3

Subtract $1024$ from $256$ .

$L = \frac{- 16 \pm \sqrt{- 768}}{2 \cdot 1}$

Step 7.2.4.1.4

Rewrite $- 768$ as $- 1 (768)$ .

$L = \frac{- 16 \pm \sqrt{- 1 \cdot 768}}{2 \cdot 1}$

Step 7.2.4.1.5

Rewrite $\sqrt{- 1 (768)}$ as $\sqrt{- 1} \cdot \sqrt{768}$ .

$L = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{768}}{2 \cdot 1}$

Step 7.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$L = \frac{- 16 \pm i \cdot \sqrt{768}}{2 \cdot 1}$

Step 7.2.4.1.7

Rewrite $768$ as $16^{2} \cdot 3$ .

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Step 7.2.4.1.7.1

Factor $256$ out of $768$ .

$L = \frac{- 16 \pm i \cdot \sqrt{256 (3)}}{2 \cdot 1}$

Step 7.2.4.1.7.2

Rewrite $256$ as $16^{2}$ .

$L = \frac{- 16 \pm i \cdot \sqrt{16^{2} \cdot 3}}{2 \cdot 1}$

Step 7.2.4.1.8

Pull terms out from under the radical.

$L = \frac{- 16 \pm i \cdot (16 \sqrt{3})}{2 \cdot 1}$

Step 7.2.4.1.9

Move $16$ to the left of $i$ .

$L = \frac{- 16 \pm 16 i \sqrt{3}}{2 \cdot 1}$

Step 7.2.4.2

Multiply $2$ by $1$ .

$L = \frac{- 16 \pm 16 i \sqrt{3}}{2}$

Step 7.2.4.3

Simplify $\frac{- 16 \pm 16 i \sqrt{3}}{2}$ .

$L = - 8 \pm 8 i \sqrt{3}$

Step 7.2.5

The final answer is the combination of both solutions.

$L = - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$

Step 8

The final solution is all the values that make $5 L (1 - \frac{L}{16}) (1 + \frac{L}{16} + \frac{L^{2}}{256}) = 0$ true.

$L = 0, 16, - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$