Basic Math Examples

3.95 \cdot 10^{- 3}

$3.95 \cdot 10^{- 3}$ ,

1.47 \cdot 10^{- 6}

$1.47 \cdot 10^{- 6}$

Step 1

The mean of a set of numbers is the sum divided by the number of terms.

¯ x = \frac{3.95 \cdot 10^{- 3} + 1.47 \cdot 10^{- 6}}{2}

$\overline{x} = \frac{3.95 \cdot 10^{- 3} + 1.47 \cdot 10^{- 6}}{2}$

Step 2

Move the decimal point in

1.47

$1.47$ to the left by

3

$3$ places and increase the power of

10^{- 6}

$10^{- 6}$ by

3

$3$ .

¯ x = \frac{3.95 \cdot 10^{- 3} + 0.00147 \cdot 10^{- 3}}{2}

$\overline{x} = \frac{3.95 \cdot 10^{- 3} + 0.00147 \cdot 10^{- 3}}{2}$

Step 3

Factor

10^{- 3}

$10^{- 3}$ out of

3.95 \cdot 10^{- 3} + 0.00147 \cdot 10^{- 3}

$3.95 \cdot 10^{- 3} + 0.00147 \cdot 10^{- 3}$ .

¯ x = \frac{(3.95 + 0.00147) \cdot 10^{- 3}}{2}

$\overline{x} = \frac{(3.95 + 0.00147) \cdot 10^{- 3}}{2}$

Step 4

Add

3.95

$3.95$ and

0.00147

$0.00147$ .

¯ x = \frac{3.95147 \cdot 10^{- 3}}{2}

$\overline{x} = \frac{3.95147 \cdot 10^{- 3}}{2}$

Step 5

Divide using scientific notation.

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Step 5.1

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

¯ x = (\frac{3.95147}{2}) (\frac{10^{- 3}}{1})

$\overline{x} = (\frac{3.95147}{2}) (\frac{10^{- 3}}{1})$

Step 5.2

Divide

3.95147

$3.95147$ by

2

$2$ .

¯ x = 1.975735 (\frac{10^{- 3}}{1})

$\overline{x} = 1.975735 (\frac{10^{- 3}}{1})$

Step 5.3

Divide

10^{- 3}

$10^{- 3}$ by

1

$1$ .

¯ x = 1.975735 \cdot 10^{- 3}

$\overline{x} = 1.975735 \cdot 10^{- 3}$

¯ x = 1.975735 \cdot 10^{- 3}

$\overline{x} = 1.975735 \cdot 10^{- 3}$

Step 6

Divide.

¯ x = 0.00197573

$\overline{x} = 0.00197573$

Step 7

Set up the formula for variance. The variance of a set of values is a measure of the spread of its values.

s^{2} = n \sum i = 1 \frac{{(x_{i} - x_{a v g})}_{i}^{2}}{n - 1}

$s^{2} = \sum_{i = 1}^{n} \frac{{(x_{i} - x_{a v g})}^{2}}{n - 1}$

Step 8

Set up the formula for variance for this set of numbers.

s = \frac{{(3.95 \cdot 10^{- 3} - 0.00197573)}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}

$s = \frac{{(3.95 \cdot 10^{- 3} - 0.00197573)}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}$

Step 9

Simplify the result.

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Step 9.1

Convert

- 0.00197573

$- 0.00197573$ to scientific notation.

s = \frac{{(3.95 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}

$s = \frac{{(3.95 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}$

Step 9.2

Factor

10^{- 3}

$10^{- 3}$ out of

3.95 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3}

$3.95 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3}$ .

s = \frac{{((3.95 - 1.975735) \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}

$s = \frac{{((3.95 - 1.975735) \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}$

Step 9.3

Subtract

1.975735

$1.975735$ from

3.95

$3.95$ .

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 0.00197573)}^{2}}{2 - 1}$

Step 9.4

Convert

- 0.00197573

$- 0.00197573$ to scientific notation.

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 1.975735 \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(1.47 \cdot 10^{- 6} - 1.975735 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.5

Move the decimal point in

1.47

$1.47$ to the left by

3

$3$ places and increase the power of

10^{- 6}

$10^{- 6}$ by

3

$3$ .

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(0.00147 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(0.00147 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.6

Simplify with factoring out.

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Step 9.6.1

Factor

10^{- 3}

$10^{- 3}$ out of

0.00147 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3}

$0.00147 \cdot 10^{- 3} - 1.975735 \cdot 10^{- 3}$ .

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {((0.00147 - 1.975735) \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {((0.00147 - 1.975735) \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.6.2

Subtract

1.975735

$1.975735$ from

0.00147

$0.00147$ .

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{(1.974265 \cdot 10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.7

Simplify the numerator.

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Step 9.7.1

Apply the product rule to

1.974265 \cdot 10^{- 3}

$1.974265 \cdot 10^{- 3}$ .

s = \frac{{1.974265}^{2} \cdot {(10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}

$s = \frac{{1.974265}^{2} \cdot {(10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.7.2

Raise

$1.974265$ to the power of

$2$ .

$s = \frac{3.89772229 \cdot {(10^{- 3})}^{2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.7.3

Multiply the exponents in

${(10^{- 3})}^{2}$ .

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Step 9.7.3.1

Apply the power rule and multiply exponents,

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

$s = \frac{3.89772229 \cdot 10^{- 3 \cdot 2} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.7.3.2

Multiply

$- 3$ by

$2$ .

$s = \frac{3.89772229 \cdot 10^{- 6} + {(- 1.974265 \cdot 10^{- 3})}^{2}}{2 - 1}$

Step 9.7.4

Apply the product rule to

$- 1.974265 \cdot 10^{- 3}$ .

$s = \frac{3.89772229 \cdot 10^{- 6} + {(- 1.974265)}^{2} \cdot {(10^{- 3})}^{2}}{2 - 1}$

Step 9.7.5

Raise

$- 1.974265$ to the power of

$2$ .

$s = \frac{3.89772229 \cdot 10^{- 6} + 3.89772229 \cdot {(10^{- 3})}^{2}}{2 - 1}$

Step 9.7.6

Multiply the exponents in

${(10^{- 3})}^{2}$ .

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Step 9.7.6.1

Apply the power rule and multiply exponents,

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

$s = \frac{3.89772229 \cdot 10^{- 6} + 3.89772229 \cdot 10^{- 3 \cdot 2}}{2 - 1}$

Step 9.7.6.2

Multiply

$- 3$ by

$2$ .

$s = \frac{3.89772229 \cdot 10^{- 6} + 3.89772229 \cdot 10^{- 6}}{2 - 1}$

Step 9.7.7

Add

$3.89772229 \cdot 10^{- 6}$ and

$3.89772229 \cdot 10^{- 6}$ .

$s = \frac{7.79544458 \cdot 10^{- 6}}{2 - 1}$

Step 9.7.8

Rewrite the expression using the negative exponent rule

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

$s = \frac{7.79544458 \cdot \frac{1}{10^{6}}}{2 - 1}$

Step 9.7.9

Raise

$10$ to the power of

$6$ .

$s = \frac{7.79544458 \cdot \frac{1}{1000000}}{2 - 1}$

Step 9.8

Subtract

$1$ from

$2$ .

$s = \frac{7.79544458 \cdot \frac{1}{1000000}}{1}$

Step 9.9

Combine

$7.79544458$ and

$\frac{1}{1000000}$ .

$s = \frac{\frac{7.79544458}{1000000}}{1}$

Step 9.10

Simplify by dividing numbers.

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Step 9.10.1

Divide

$7.79544458$ by

$1000000$ .

$s = \frac{0.00000779}{1}$

Step 9.10.2

Divide

$0.00000779$ by

$1$ .

$s = 0.00000779$

Step 10

Approximate the result.

$s^{2} \approx 0$