Problem 59 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. A ribosome of \(\mathbf{E}\). coli has a mass of about \(4.7 \times 10^{-19}\) grams.

Step-by-Step Solution

Answer

Question: Convert the mass of a ribosome of E. coli given in scientific notation \(4.7 \times 10^{-19}\) grams to standard decimal form. Answer: The mass of a ribosome of E. coli in standard decimal form is approximately \(0.00000000000000000047\) grams.

Step 1: Identify the scientific notation

The given mass of ribosome in scientific notation is \(4.7 \times 10^{-19}\) grams.

Step 2: Shift the decimal point

Since -19 is the exponent in our scientific notation, we will move the decimal point 19 places to the left. Add zeros as placeholders for those empty decimal places.

Step 3: Convert the scientific notation to decimal form

After shifting the decimal point 19 places to the left, the number in standard decimal form is \(0.00000000000000000047\) grams. So, the mass of a ribosome of E. coli in standard decimal form is approximately \(0.00000000000000000047\) grams.

Key Concepts

Decimal Conversion

Understanding decimal conversion is crucial when reading numbers in scientific notation and translating them into standard form. Scientific notation is often used in mathematics and science to represent very large or very small numbers in a compact format. It looks something like this: a product of a number (usually between 1 and 10) and a power of ten. For instance, in the exercise, the ribosome mass is given as \(4.7 \times 10^{-19}\).

To convert from scientific notation to a decimal, you'll have to shift the decimal point. Here’s a simple way to break it down:

• Identify the exponent in the scientific notation. The exponent in our example is -19.
• Move the decimal point in the base number (4.7) in the direction and number of places indicated by the exponent.
• For negative exponents, like our -19, shift the decimal point to the left. For positive exponents, move the decimal point to the right.
• Use zeros to fill in any spaces when you move the decimal point.

Applying these rules, shifting \(4.7\) nineteen places to the left fills in with zeros, resulting in \(0.00000000000000000047\). Thus, converting scientific notation to decimal isn't just about numbers; it involves understanding place value and adjusting accordingly.

Exponents

Exponents are a mathematical notation showing how many times a number, called the base, is multiplied by itself. When you see a number like \(10^{-19}\), the exponent is \(-19\). This tells us two things about the number:

• The base is 10. This is common in our decimal system and powers of ten are essential for representing large and small numbers.
• The exponent is -19, which indicates that the base 10 should be used in the denominator of a fraction in standard decimal form.

Understanding Negative Exponents
Negative exponents represent a fraction with 1 as the numerator and the base raised to the absolute value of the exponent as the denominator. For example:

\[10^{-19} = \frac{1}{10^{19}}\]

Why Use Exponents in Scientific Notation?

In scientific notation, exponents simplify writing and calculating with extremely large or tiny numbers. Instead of writing many zeros, we use an exponent to show how many zeros would typically follow the leading digits. This makes it easier for scientists and mathematicians to read and understand these vast quantities. Understanding this allows you to deftly navigate and manipulate similarly styled numbers in real-world scientific contexts.

Mathematics Education

Gaining a solid grasp of concepts like decimal conversion and exponents is an important component of mathematics education. These skills foster critical thinking and numerical literacy, which are foundational for understanding more complex topics later in your studies. Here's how these concepts fit into a broader educational context:

Why Is This Important?
The purpose of learning these topics is not just about doing well in exams. Practically, understanding exponents and converting decimals enables students to:

• Work with large and small numbers succinctly, both in academic work and everyday life.
• Develop problem-solving abilities by visualizing and contextualizing problems more effectively.
• Prepare for advanced topics in subjects like chemistry, physics, and engineering, where these concepts are commonly applied.

Building Strong Numerical Skills
Math education empowers you to make sense of data, draw logical conclusions, and assess information critically. With technologies such as calculators and spreadsheets often automating these calculations, understanding the underlying principles such as exponents and decimal conversion remains essential. It is this fundamental knowledge that allows students to develop precision and confidence in numerical reasoning and application. Keep practicing, stay curious, and seek clarity when concepts seem challenging!