Chapter 2: Problem 40
Write the following measurements in scientific notation. $$\begin{array}{l}{\text { a. } 800000000 \mathrm{m}} \\ {\text { b. } 0.00095 \mathrm{m}} \\ {\text { c. } 60200 \mathrm{L}} \\ {\text { d. } 0.0015 \mathrm{kg}}\end{array}$$
Short Answer
Expert verified
The numbers in scientific notation are: a) \(8\times 10^8\) m, b) \(9.5\times 10^{-4}\) m, c) \(6.02\times 10^4\) L, and d) \(1.5\times 10^{-3}\) kg.
Step by step solution
01
Convert 800000000 m to scientific notation
We can write the number as \(8\times 10^n\). Now calculate \(n\) using the fact that we have 9 digits in the number, which means that \(n = 8\). Therefore, the number in scientific notation is \(8\times 10^8\) m.
02
Convert 0.00095 m to scientific notation
Again, we can write the number as \(9.5\times 10^n\). Now, to find \(n\), refer to after the decimal point where the first non-zero digit appears, which in this case appears after 4 decimal places. Therefore, our \(n\) is -4. Thus, the number in scientific notation is \(9.5\times 10^{-4}\) m.
03
Convert 60200 L to scientific notation
The number can be represented as \(6.02\times 10^n\). To find \(n\), count the number of digits after the first digit, which in our case is 4. Hence, the number is \(6.02\times 10^4\) L.
04
Convert 0.0015 kg to scientific notation
The number can be represented as \(1.5\times 10^n\). To find \(n\), look where the first non-zero number appears after the decimal point, which occurs after 3 decimal places. So, \(n = -3\), and the number in scientific notation is \(1.5\times 10^{-3}\) kg.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Notational Conversion
When dealing with very large or very small numbers, it's often useful to convert them into scientific notation for ease of use in calculations and to clearly display the magnitude of the number. Notational conversion is the process of translating a number into scientific notation, which expresses a number as the product of a coefficient (usually between 1 and 10) and a power of 10. This method is particularly useful in science and engineering to simplify measurements and computations.
For instance, let's consider the number 60200 L. In standard notation, it might be challenging to quickly gauge its size, but converting it to scientific notation as
Steps involved in notational conversion typically include:
For instance, let's consider the number 60200 L. In standard notation, it might be challenging to quickly gauge its size, but converting it to scientific notation as
\(6.02 \times 10^4\text{L}\)
immediately gives us an understanding that the number is in the ten-thousands.Steps involved in notational conversion typically include:
- Identifying the first non-zero digit in the number.
- Placing a decimal point after this digit to establish the coefficient.
- Counting the number of places the decimal has moved to determine the exponent.
- Writing the converted number as a product of this coefficient and 10 raised to the necessary power.
Exponential Representation
Exponential representation is a way of expressing numbers using exponents, which indicate how many times a number (the base) is multiplied by itself. Scientific notation is a specific form of exponential representation where the base is always 10. This notation is highly efficient for representing very large numbers, such as the distance between stars, or very small numbers, like the mass of an atom.
Here's how we utilize exponential representation in scientific notation:
Understanding exponential representation is a powerful tool, simplifying not only the expression of numbers but also the manipulation of them in mathematical operations.
Here's how we utilize exponential representation in scientific notation:
- The coefficient must be a number between 1 and 10.
- The exponent on the base 10 indicates the number of decimal places the decimal point would move in standard notation.
- A positive exponent shows that the decimal moves to the right for large numbers.
- A negative exponent indicates that the decimal moves to the left for small numbers.
\(1.5 \times 10^{-3}\text{kg}\)
.Understanding exponential representation is a powerful tool, simplifying not only the expression of numbers but also the manipulation of them in mathematical operations.
Scientific Notation Practice
Scientific notation practice involves regularly working with this system of writing numbers to become comfortable with converting to and from standard form, performing arithmetic operations, and recognizing how scientific notation helps to conceptualize different orders of magnitude. Let's apply this practice to a few examples.
For the number 800000000 m, practicing the conversion steps yields a scientific notation of
Similarly, with a very small number like 0.00095 m, practice helps you quickly recognize that the first significant digit is 9, requiring the decimal to move four places to the left. Thus, the scientific notation is
To foster competency in scientific notation, engage in exercises that span a range of magnitudes and include both multiplication and division problems. Efficiently converting and operating with these numbers can significantly streamline the problem-solving process in various scientific disciplines.
For the number 800000000 m, practicing the conversion steps yields a scientific notation of
\(8 \times 10^8 \text{m}\)
. Remember, the skill lies in placing the decimal after the first non-zero digit and counting how many places the decimal point has been moved to determine the exponent. This number's conversion shows an 8 followed by eight zeroes, resulting in an exponent of 8.Similarly, with a very small number like 0.00095 m, practice helps you quickly recognize that the first significant digit is 9, requiring the decimal to move four places to the left. Thus, the scientific notation is
\(9.5 \times 10^{-4}\text{m}\)
, showing our confidence with a negative exponent.To foster competency in scientific notation, engage in exercises that span a range of magnitudes and include both multiplication and division problems. Efficiently converting and operating with these numbers can significantly streamline the problem-solving process in various scientific disciplines.