Question

Write the following numbers in scientific notation. \(\begin{array}{ll}{\text { a. } 0.0045834 \mathrm{mm}} & {\text { c. } 438,904 \mathrm{s}} \\ {\text { b. } 0.03054 \mathrm{g}} & {\text { d. } 7,004,300,000 \mathrm{g}}\end{array}\)

Short answer

a. \(4.5834 \times 10^{-3}\) mm b. \(3.054 \times 10^{-2}\) g c. \(4.38904 \times 10^{5}\) s d. \(7.0043 \times 10^{9}\) g

Step-by-step solution

Understand scientific notation

Scientific notation is a way to express very large or very small numbers in a more convenient and compact form. It is written in the format \(A \times 10^{B}\), where A (the coefficient) is a number greater than or equal to 1 and less than 10, and B (the exponent) is an integer that represents the power of 10.

Write 0.0045834 in scientific notation

In order to write the number 0.0045834 in scientific notation, we can follow these steps: 1. Move the decimal point to the right until you have a number between 1 and 10. In this case, the number becomes 4.5834. 2. Count the number of times you moved the decimal point to the right. Here, we moved the decimal point 3 times. 3. Write the number in the format \(A \times 10^{B}\). In this case, A = 4.5834 and B = -3 because we moved the decimal point 3 times to the right. So, the number in scientific notation is \(4.5834 \times 10^{-3}\).

Write 0.03054 in scientific notation

Following the same steps for the number 0.03054: 1. The number becomes 3.054 after moving the decimal point 2 times to the right. 2. We moved the decimal point 2 times. 3. The number in scientific notation is \(3.054 \times 10^{-2}\).

Write 438,904 in scientific notation

For the number 438,904: 1. Move the decimal point to the left until you have a number between 1 and 10. The number becomes 4.38904 2. Count the number of times you moved the decimal point to the left. In this case, we moved it 5 times. 3. Write the number in scientific notation as \(4.38904 \times 10^{5}\).

Write 7,004,300,000 in scientific notation

For the number 7,004,300,000: 1. The number becomes 7.0043 after moving the decimal point 9 times to the left. 2. We moved the decimal point 9 times. 3. The number in scientific notation is \(7.0043 \times 10^{9}\).

Combine the scientific notation with their units

Now we can combine each number in scientific notation with their corresponding units: a. \(4.5834 \times 10^{-3}\) mm b. \(3.054 \times 10^{-2}\) g c. \(4.38904 \times 10^{5}\) s d. \(7.0043 \times 10^{9}\) g

Key concepts

Decimal Point Movement

Understanding how to manipulate the decimal point is crucial when converting numbers into scientific notation. The basis of scientific notation relies on shifting the decimal point to create a simpler form of the original number.
For small numbers, like 0.0045834, you move the decimal point to the right to form a number between 1 and 10, resulting in 4.5834. Count the shifts — here, it’s 3 — and represent this movement as a negative exponent: -3.
For large numbers, such as 7,004,300,000, the process reverses. Move the decimal point to the left to obtain a number between 1 and 10, which becomes 7.0043. Again, count the moves — 9 in this instance — and the exponent is positive: 9.
Using these methods to align the decimal for simple numbers eases the conversion into scientific notation efficiently. Always keep in mind whether the number was originally small or large to determine if the exponent will be positive or negative.

Exponentiation

Exponentiation in scientific notation involves expressing the number of times a decimal point is shifted as an exponent. This form accelerates calculations with very large or very small numbers, streamlining both understanding and solving processes.
For the number 0.03054, moving the decimal point 2 times to the right yields a coefficient of 3.054. This shift is denoted by the negative exponent -2 because the original number was less than 1. Hence, it is written as:

• \(3.054 \times 10^{-2}\)

Similarly, the number 438,904 shifts the decimal point 5 times to the left, forming

• \(4.38904 \times 10^{5}\).

This positive exponent represents the large size of the original number.
The exponent indicates whether your initial number was tiny or huge, aiding in understanding the magnitude while ensuring precise calculations across mathematical contexts.

Unit Conversion

When working with numbers in scientific notation, it’s practical to retain the original units paired with each value. Scientific notation highlights the number’s magnitude but can sometimes obscure the associated measurement unit if not handled correctly.
In the exercise, units such as millimeters (mm), grams (g), and seconds (s) are part of each number’s context. Using scientific notation doesn't change these units; rather, it helps present the magnitude in a more digestible format:

• \(4.5834 \times 10^{-3}\) mm, emphasizing its minuteness.
• \(7.0043 \times 10^{9}\) g, indicating an enormous quantity.

Always append the correct unit to provide context and maintain the integrity of measurements.
This association of converted numbers to their original units ensures comprehension and applicability, supporting precise contextual meaning in varied scientific or mathematical discussions.