Calculus Examples

$10.4 \cdot 10^{- 3} v - \frac{6.14 \cdot 10^{8}}{v^{3}} = 0$

Step 1

Move the decimal point in $10.4$ to the left by $1$ place and increase the power of $10^{- 3}$ by $1$ .

$1.04 \cdot 10^{- 2} v - \frac{6.14 \cdot 10^{8}}{v^{3}} = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$1, 1, v^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$v^{3}$

Step 3

Multiply each term in $1.04 \cdot 10^{- 2} v - \frac{6.14 \cdot 10^{8}}{v^{3}} = 0$ by $v^{3}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $1.04 \cdot 10^{- 2} v - \frac{6.14 \cdot 10^{8}}{v^{3}} = 0$ by $v^{3}$ .

$1.04 \cdot 10^{- 2} v \cdot v^{3} - \frac{6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $v$ by $v^{3}$ by adding the exponents.

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Step 3.2.1.1.1

Move $v^{3}$ .

$1.04 \cdot 10^{- 2} (v^{3} v) - \frac{6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.1.2

Multiply $v^{3}$ by $v$ .

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Step 3.2.1.1.2.1

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

$1.04 \cdot 10^{- 2} (v^{3} v^{1}) - \frac{6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.1.2.2

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

$1.04 \cdot 10^{- 2} v^{3 + 1} - \frac{6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.1.3

Add $3$ and $1$ .

$1.04 \cdot 10^{- 2} v^{4} - \frac{6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.2

Cancel the common factor of $v^{3}$ .

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Step 3.2.1.2.1

Move the leading negative in $- \frac{6.14 \cdot 10^{8}}{v^{3}}$ into the numerator.

$1.04 \cdot 10^{- 2} v^{4} + \frac{- 6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.2.2

Cancel the common factor.

$1.04 \cdot 10^{- 2} v^{4} + \frac{- 6.14 \cdot 10^{8}}{v^{3}} v^{3} = 0 v^{3}$

Step 3.2.1.2.3

Rewrite the expression.

$1.04 \cdot 10^{- 2} v^{4} - 6.14 \cdot 10^{8} = 0 v^{3}$

Step 3.2.2

Reorder factors in $1.04 \cdot 10^{- 2} v^{4} - 6.14 \cdot 10^{8}$ .

$1.04 v^{4} \cdot 10^{- 2} - 6.14 \cdot 10^{8} = 0 v^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $0$ by $v^{3}$ .

$1.04 v^{4} \cdot 10^{- 2} - 6.14 \cdot 10^{8} = 0$

Step 4

Solve the equation.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1.04 v^{4} \cdot \frac{1}{10^{2}} - 6.14 \cdot 10^{8} = 0$

Step 4.1.2

Raise $10$ to the power of $2$ .

$1.04 v^{4} \cdot \frac{1}{100} - 6.14 \cdot 10^{8} = 0$

Step 4.1.3

Multiply $1.04 v^{4} \frac{1}{100}$ .

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Step 4.1.3.1

Combine $1.04$ and $\frac{1}{100}$ .

$v^{4} \frac{1.04}{100} - 6.14 \cdot 10^{8} = 0$

Step 4.1.3.2

Combine $v^{4}$ and $\frac{1.04}{100}$ .

$\frac{v^{4} \cdot 1.04}{100} - 6.14 \cdot 10^{8} = 0$

Step 4.1.4

Move $1.04$ to the left of $v^{4}$ .

$\frac{1.04 \cdot v^{4}}{100} - 6.14 \cdot 10^{8} = 0$

Step 4.1.5

Factor $1.04$ out of $1.04 \cdot v^{4}$ .

$\frac{1.04 (v^{4})}{100} - 6.14 \cdot 10^{8} = 0$

Step 4.1.6

Factor $100$ out of $100$ .

$\frac{1.04 (v^{4})}{100 (1)} - 6.14 \cdot 10^{8} = 0$

Step 4.1.7

Separate fractions.

$\frac{1.04}{100} \cdot \frac{v^{4}}{1} - 6.14 \cdot 10^{8} = 0$

Step 4.1.8

Divide $1.04$ by $100$ .

$0.0104 \frac{v^{4}}{1} - 6.14 \cdot 10^{8} = 0$

Step 4.1.9

Divide $v^{4}$ by $1$ .

$0.0104 v^{4} - 6.14 \cdot 10^{8} = 0$

Step 4.2

Add $6.14 \cdot 10^{8}$ to both sides of the equation.

$0.0104 v^{4} = 6.14 \cdot 10^{8}$

Step 4.3

Divide each term in $0.0104 v^{4} = 6.14 \cdot 10^{8}$ by $0.0104$ and simplify.

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Step 4.3.1

Divide each term in $0.0104 v^{4} = 6.14 \cdot 10^{8}$ by $0.0104$ .

$\frac{0.0104 v^{4}}{0.0104} = \frac{6.14 \cdot 10^{8}}{0.0104}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $0.0104$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{0.0104 v^{4}}{0.0104} = \frac{6.14 \cdot 10^{8}}{0.0104}$

Step 4.3.2.1.2

Divide $v^{4}$ by $1$ .

$v^{4} = \frac{6.14 \cdot 10^{8}}{0.0104}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide using scientific notation.

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Step 4.3.3.1.1

Group coefficients together and exponents together to divide numbers in scientific notation.

$v^{4} = (\frac{6.14}{0.0104}) (\frac{10^{8}}{1})$

Step 4.3.3.1.2

Divide $6.14$ by $0.0104$ .

$v^{4} = 590.\overline{384615} \frac{10^{8}}{1}$

Step 4.3.3.1.3

Divide $10^{8}$ by $1$ .

$v^{4} = 590.\overline{384615} \cdot 10^{8}$

Step 4.3.3.2

Move the decimal point in $590.\overline{384615}$ to the left by $2$ places and increase the power of $10^{8}$ by $2$ .

$v^{4} = 5.90384615 \cdot 10^{10}$

Step 4.4

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

$v = \pm \sqrt[4]{5.90384615 \cdot 10^{10}}$

Step 4.5

Simplify $\pm \sqrt[4]{5.90384615 \cdot 10^{10}}$ .

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Step 4.5.1

Rewrite $5.90384615 \cdot 10^{10}$ as $0.05903846 \cdot 10^{12}$ .

$v = \pm \sqrt[4]{0.05903846 \cdot 10^{12}}$

Step 4.5.2

Rewrite $\sqrt[4]{0.05903846 \cdot 10^{12}}$ as $\sqrt[4]{0.05903846} \cdot \sqrt[4]{10^{12}}$ .

$v = \pm \sqrt[4]{0.05903846} \cdot \sqrt[4]{10^{12}}$

Step 4.5.3

Evaluate the root.

$v = \pm 0.4929283 \cdot \sqrt[4]{10^{12}}$

Step 4.5.4

Rewrite $10^{12}$ as ${(10^{3})}^{4}$ .

$v = \pm 0.4929283 \cdot \sqrt[4]{{(10^{3})}^{4}}$

Step 4.5.5

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

$v = \pm 0.4929283 \cdot 10^{3}$

Step 4.5.6

Move the decimal point in $0.4929283$ to the right by $1$ place and decrease the power of $10^{3}$ by $1$ .

$v = \pm 4.92928306 \cdot 10^{2}$

Step 4.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.6.1

First, use the positive value of the $\pm$ to find the first solution.

$v = 4.92928306 \cdot 10^{2}$

Step 4.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$v = - 4.92928306 \cdot 10^{2}$

Step 4.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$v = 4.92928306 \cdot 10^{2}, - 4.92928306 \cdot 10^{2}$

Step 5

The result can be shown in multiple forms.

Scientific Notation:

$v = 4.92928306 \cdot 10^{2}, - 4.92928306 \cdot 10^{2}$

Expanded Form:

$v = 492.9283061, - 492.9283061$