Problem 52 - Elementary Algebra

Question

For the following problems, convert the numbers from scientific notation to standard decimal form. The mass of the earth is about \(5.98 \times 10^{27}\) grams.

Step-by-Step Solution

Answer

Answer: The mass of the Earth in standard decimal form is approximately 5,980,000,000,000,000,000,000,000,000 grams.

Step 1: Identify the number in scientific notation

We are given the mass of the earth in scientific notation, which is \(5.98 \times 10^{27}\) grams.

Step 2: Move the decimal point

To convert the number to its standard decimal form, move the decimal point 27 places to the right, as indicated by the exponent 27.

Step 3: Fill with zeroes

Since we moved the decimal point 27 places to the right, we need to fill in with zeroes for any empty spaces created in the process: \(5.98 \times 10^{27} = 5,980,000,000,000,000,000,000,000,000\) grams

Step 4: Write the final answer

The mass of the earth in standard decimal form is approximately 5,980,000,000,000,000,000,000,000,000 grams.

Key Concepts

Decimal Conversion

Decimal conversion involves changing a number from one form to another, such as from scientific notation to standard form. In this exercise, we have the number expressed in scientific notation: \(5.98 \times 10^{27}\). This notation is useful for writing very large or very small numbers in a compact form. To convert it, you primarily need to move the decimal point. Here, \(10^{27}\) tells you to move the decimal right by 27 places.

• Identify the exponents to understand how many positions to move.
• Shift the decimal point the specified number of places (to the right for a positive exponent).
• Add zeroes to fill any gaps created by the movement.

This process transforms the scientific notation to a standard decimal format, making it easier to read and understand in everyday usage.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. Here, our base is 10, and our exponent is 27. This indicates how many times the base (10) multiplies itself. Therefore, \(10^{27}\) equals a 1 followed by 27 zeros.
When numbers are coupled with exponentiation in scientific notation, it allows for easy expression and manipulation of significantly large values:

• Think of "\(10^{27}\)" as presenting the size or scale of the number.
• Recognize exponentiation used to simplify the notation of massive or microscopic quantities.

Exponentiation underpins the functionality of scientific notation, providing a succinct way to display numbers like Earth's mass.

Standard Form

The standard form is a way of writing numbers in their full numerically expanded version. It is utilized for accessibility and comprehensibility, especially when dealing with extremely large or small quantities. After converting the scientific notation of the Earth's mass into standard form, we get \(5,980,000,000,000,000,000,000,000,000\) grams.
To achieve this:

• Begin with the original number from scientific notation \(5.98 \times 10^{27}\).
• Use decimal conversion and exponentiation understanding.
• Result in an easy-to-read full numerical representation.

Representing numbers like the Earth's mass in standard form offers clarity, making them visually and mentally manageable in practical contexts.