Calculus Examples

$\frac{1}{2} \cdot 9.1 \cdot 10^{- 31} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1

Simplify.

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Step 1.1

Multiply

$\frac{1}{2} \cdot 9.1 \cdot 10^{- 31}$ .

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Step 1.1.1

Combine

$9.1$ and

$\frac{1}{2}$ .

$\frac{9.1}{2} \cdot 10^{- 31} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1.1.2

Combine

$\frac{9.1}{2}$ and

$10^{- 31}$ .

$\frac{9.1 \cdot 10^{- 31}}{2} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1.2

Divide using scientific notation.

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Step 1.2.1

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

$(\frac{9.1}{2}) (\frac{10^{- 31}}{1}) \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1.2.2

Divide

$9.1$ by

$2$ .

$4.55 \frac{10^{- 31}}{1} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1.2.3

Divide

$10^{- 31}$ by

$1$ .

$4.55 \cdot 10^{- 31} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 175 \cdot 10^{6}$

Step 1.3

Move the decimal point in

$175$ to the left by

$2$ places and increase the power of

$10^{6}$ by

$2$ .

$4.55 \cdot 10^{- 31} \cdot v^{2} = 1.6 \cdot 10^{- 19} \cdot 1.75 \cdot 10^{8}$

Step 1.4

Multiply

$1.6$ by

$1.75$ .

$4.55 \cdot 10^{- 31} \cdot v^{2} = 2.8 (10^{- 19} \cdot 10^{8})$

Step 1.5

Multiply

$10^{- 19}$ by

$10^{8}$ by adding the exponents.

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Step 1.5.1

Use the power rule

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

$4.55 \cdot 10^{- 31} \cdot v^{2} = 2.8 \cdot 10^{- 19 + 8}$

Step 1.5.2

Add

$- 19$ and

$8$ .

$4.55 \cdot 10^{- 31} \cdot v^{2} = 2.8 \cdot 10^{- 11}$

Step 2

Divide each term in

$4.55 \cdot 10^{- 31} \cdot v^{2} = 2.8 \cdot 10^{- 11}$ by

$4.55 \cdot 10^{- 31}$ and simplify.

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Step 2.1

Divide each term in

$4.55 \cdot 10^{- 31} \cdot v^{2} = 2.8 \cdot 10^{- 11}$ by

$4.55 \cdot 10^{- 31}$ .

$\frac{4.55 \cdot 10^{- 31} \cdot v^{2}}{4.55 \cdot 10^{- 31}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

$4.55$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4.55 \cdot 10^{- 31} \cdot v^{2}}{4.55 \cdot 10^{- 31}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.1.2

Rewrite the expression.

$\frac{10^{- 31} \cdot v^{2}}{10^{- 31}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.2

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{10^{31}} v^{2}}{10^{- 31}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.3

Rewrite the expression using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{10^{31}} v^{2}}{\frac{1}{10^{31}}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.4

Combine

$\frac{1}{10^{31}}$ and

$v^{2}$ .

$\frac{\frac{v^{2}}{10^{31}}}{\frac{1}{10^{31}}} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{v^{2}}{10^{31}} \cdot 10^{31} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.6

Cancel the common factor of

$10^{31}$ .

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Step 2.2.6.1

Cancel the common factor.

$\frac{v^{2}}{10^{31}} \cdot 10^{31} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.2.6.2

Rewrite the expression.

$v^{2} = \frac{2.8 \cdot 10^{- 11}}{4.55 \cdot 10^{- 31}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide using scientific notation.

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Step 2.3.1.1

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

$v^{2} = (\frac{2.8}{4.55}) (\frac{10^{- 11}}{10^{- 31}})$

Step 2.3.1.2

Divide

$2.8$ by

$4.55$ .

$v^{2} = 0.\overline{615384} \frac{10^{- 11}}{10^{- 31}}$

Step 2.3.1.3

Subtract the exponent from the denominator from the exponent of the numerator for the same base

$v^{2} = 0.\overline{615384} \cdot 10^{- 11 - 31 \cdot - 1}$

Step 2.3.1.4

Multiply

$- 31$ by

$- 1$ .

$v^{2} = 0.\overline{615384} \cdot 10^{- 11 + 31}$

Step 2.3.1.5

Add

$- 11$ and

$31$ .

$v^{2} = 0.\overline{615384} \cdot 10^{20}$

Step 2.3.2

Move the decimal point in

$0.\overline{615384}$ to the right by

$1$ place and decrease the power of

$10^{20}$ by

$1$ .

$v^{2} = 6.\overline{153846} \cdot 10^{19}$

Step 3

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

$v = \pm \sqrt{6.\overline{153846} \cdot 10^{19}}$

Step 4

Simplify

$\pm \sqrt{6.\overline{153846} \cdot 10^{19}}$ .

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Step 4.1

Rewrite

$6.\overline{153846} \cdot 10^{19}$ as

$0.\overline{615384} \cdot 10^{20}$ .

$v = \pm \sqrt{0.\overline{615384} \cdot 10^{20}}$

Step 4.2

Rewrite

$\sqrt{0.\overline{615384} \cdot 10^{20}}$ as

$\sqrt{0.\overline{615384}} \cdot \sqrt{10^{20}}$ .

$v = \pm \sqrt{0.\overline{615384}} \cdot \sqrt{10^{20}}$

Step 4.3

Evaluate the root.

$v = \pm 0.78446454 \cdot \sqrt{10^{20}}$

Step 4.4

Rewrite

$10^{20}$ as

${(10^{10})}^{2}$ .

$v = \pm 0.78446454 \cdot \sqrt{{(10^{10})}^{2}}$

Step 4.5

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

$v = \pm 0.78446454 \cdot 10^{10}$

Step 4.6

Move the decimal point in

$0.78446454$ to the right by

$1$ place and decrease the power of

$10^{10}$ by

$1$ .

$v = \pm 7.8446454 \cdot 10^{9}$

Step 5

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

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Step 5.1

First, use the positive value of the

$\pm$ to find the first solution.

$v = 7.8446454 \cdot 10^{9}$

Step 5.2

Next, use the negative value of the

$\pm$ to find the second solution.

$v = - 7.8446454 \cdot 10^{9}$

Step 5.3

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

$v = 7.8446454 \cdot 10^{9}, - 7.8446454 \cdot 10^{9}$

Step 6

The result can be shown in multiple forms.

Scientific Notation:

$v = 7.8446454 \cdot 10^{9}, - 7.8446454 \cdot 10^{9}$

Expanded Form:

$v = 7844645405.52736, - 7844645405.52736$