Problem 77 - Algebra and Trigonometry

Question

\(73-80\) . Write each number in scientific notation. $$ 129,540,000 $$

Step-by-Step Solution

Answer

The scientific notation of 129,540,000 is \( 1.2954 \times 10^8 \).

Step 1: Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. A number is written in scientific notation if it is in the form of \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Identify the Significant Figures

Find the significant figures in your number, which is 129,540,000. In this case, they are 1, 2, 9, 5, and 4.

Step 3: Create a Decimal Number Between 1 and 10

Place a decimal point to create a number between 1 and 10 using the significant figures. For 129,540,000, that number is 1.2954.

Step 4: Count the Zeros and Digits to Set the Exponent

Count how many places you move the decimal point to get from 129,540,000 to 1.2954. Move it 8 places to the left, meaning the exponent is 8.

Step 5: Write in Scientific Notation

Combine the number from Step 3 with the exponent from Step 4 to write the scientific notation: \( 1.2954 \times 10^8 \).

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its accuracy. They encompass all the non-zero digits, as well as any zeros that are sandwiched between them, or that are trailing after a decimal point. These figures are crucial while converting numbers into scientific notation because they ensure the number is precise and meaningful.

• For instance, in 129,540,000, the significant figures are 1, 2, 9, 5, and 4. They represent the precision of the original number.
• By focusing on these figures, you can obtain an approximate form of the number that retains its critical details.

Understanding how significant figures operate helps us in ensuring that the number we use in scientific notation still reflects the level of accuracy present in the original number.

Exponent

The exponent in scientific notation plays a key role as it indicates the power of ten by which the base number is multiplied. This exponent tells us how many places we need to move the decimal point to retrieve the original number from its scientific notation.
Imagine you have moved the decimal point from the number 129,540,000 to form 1.2954. To turn 1.2954 back into the original number, you moved the decimal 8 places to the left. This means your exponent is 8.

• In our example, we write it as \(10^8\).
• The positive exponent signifies a large number, confirming its vastness over the set imaginary zero point.

This understanding of exponents allows you to navigate between the condensed form of numbers in scientific notation and their true magnitude effectively.

Decimal Point

The decimal point is a small yet mighty tool in number representation, essential in both everyday math and scientific notation. It serves as the divider between the integer and fractional components of a number. By shifting the decimal point, we can represent large or small numbers more compactly.
In scientific notation, you position the decimal point immediately after the first significant figure. This ensures the resulting number is between 1 and 10, which is a key requirement of scientific notation.

• For the number 129,540,000, moving the decimal after the first number means you end up with 1.2954.
• It guides the accurate placement of exponents to display the real-value scales and sizes correctly.

Hence, mastering the role of the decimal point enables you to comprehend and apply scientific notation expertly, turning complex computations into simple, manageable expressions.