Answered

Complete the following conversions between units. All answers should be in scientific notation and have proper significant digits.



$\begin{array}{l}

9.102 \text{ MW} = 9.102 \times 10^{\text{______}} \text{ W} \\

193 \text{ mg} = 1.93 \times 10^{\text{______}} \text{ g} \\

308 \ \mu\text{L} = 3.08 \times 10^{\text{______}} \text{ L} \\

13.9 \text{ m} = 1.39 \times 10^{\text{______}} \text{ cm} \\

1.68 \text{ mol} = 1.68 \times 10^{\text{______}} \text{ mmol}

\end{array}$

Verified Answer :

  • For \(9.102 \text{ MW}\), apply the multiplier \(10^6\), resulting in \(9.102 \times 10^6\).
  • For \(193 \text{ mg}\), convert to \(1.93 \times 10^2\) and multiply by the prefix \(10^{-3}\), resulting in \(1.93 \times 10^{-1}\).
  • For \(308 \, \mu\text{L}\), convert to \(3.08 \times 10^2\) and multiply by the prefix \(10^{-6}\), resulting in \(3.08 \times 10^{-4}\).
  • For \(13.9 \text{ m}\) and \(1.68 \text{ mol}\), adjust the decimal place and multiply by \(10^1\) and \(10^0\) respectively.
  • The final exponents are \(\boxed{6, -1, -4, 1, 0}\).

Explanation

  1. Data and Analysis

The objective is to convert various physical quantities into scientific notation, specifically in the format \(a \times 10^n\), where \(1 \le |a| < 10\). We must account for metric prefixes and maintain correct significant digits. The prefixes are:

  • Mega (M) \(= 10^6\)
  • milli (m) \(= 10^{-3}\)
  • micro (\(\mu\)) \(= 10^{-6}\)

The values provided are:

  1. \(9.102 \text{ MW}\)
  2. \(193 \text{ mg}\)
  3. \(308 \, \mu\text{L}\)
  4. \(13.9 \text{ m}\)
  5. \(1.68 \text{ mol}\)
  1. Calculations for Each Conversion
  2. For \(9.102 \text{ MW}\):

Since \(1 \text{ MW} = 10^6 \text{ W}\), then \(9.102 \text{ MW} = 9.102 \times 10^6 \text{ W}\). Thus, \(n = 6\).

  1. For \(193 \text{ mg}\):

First, express \(193\) in scientific notation: \(193 = 1.93 \times 10^2\).
Since \(1 \text{ mg} = 10^{-3} \text{ g}\), then \(193 \text{ mg} = 1.93 \times 10^2 \times 10^{-3} \text{ g} = 1.93 \times 10^{2-3} \text{ g} = 1.93 \times 10^{-1} \text{ g}\). Thus, \(n = -1\).

  1. For \(308 \, \mu\text{L}\):

First, express \(308\) in scientific notation: \(308 = 3.08 \times 10^2\).
Since \(1 \, \mu\text{L} = 10^{-6} \text{ L}\), then \(308 \, \mu\text{L} = 3.08 \times 10^2 \times 10^{-6} \text{ L} = 3.08 \times 10^{2-6} \text{ L} = 3.08 \times 10^{-4} \text{ L}\). Thus, \(n = -4\).

  1. For \(13.9 \text{ m}\):

Express \(13.9\) in scientific notation: \(13.9 = 1.39 \times 10^1\). Thus, \(n = 1\).

  1. For \(1.68 \text{ mol}\):

The coefficient \(1.68\) is already in the range \(1 \le |a| < 10\). Therefore, we multiply by \(10^0\) (since \(10^0 = 1\)). Thus, \(n = 0\).

  1. Conclusion

The exponents \(n\) for each conversion are:

  • \(9.102 \times 10^6\)
  • \(1.93 \times 10^{-1}\)
  • \(3.08 \times 10^{-4}\)
  • \(1.39 \times 10^1\)
  • \(1.68 \times 10^0\)

The final answers are \(n = 6, -1, -4, 1, 0\) respectively.

Examples

Scientific notation is widely used by scientists and engineers to express very large or very small numbers compactly. For instance, in geology, the mass of the Earth is approximately \(5.97 \times 10^{24} \text{ kg}\), and in microbiology, the length of a typical bacterium might be \(2.0 \times 10^{-6} \text{ m}\). Understanding these conversions allows for easier calculations and better clarity when working with measurements across different scales, ensuring that precision and magnitude are accurately represented in technical documents.

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