Problem 116 - Elementary Algebra

Question

Problem: Evaluate \(\left(6.4 \times 10^{5}\right)\left(3 \times 10^{8}\right)\). Write the answer in scientific notation. Incorrect Answer: \(19.2 \times 10^{13}\)

Step-by-Step Solution

Answer

The correct answer is \(1.92 \times 10^{14}\) in scientific notation.

Step 1: Multiply the Significant Figures

First, multiply the significant figures of the two numbers: \(6.4\) and \(3\). \[6.4 \times 3 = 19.2\]

Step 2: Add the Exponents

Add the exponents of the two powers of ten: \[5 + 8 = 13\]

Step 3: Combine the Results

Combine the results from Steps 1 and 2: \[19.2 \times 10^{13}\]

Step 4: Normalize the Scientific Notation

Convert \(19.2 \times 10^{13}\) to proper scientific notation by recognizing that \(19.2\) can be written as \(1.92 \times 10^1\). Thus, \[19.2 \times 10^{13} = 1.92 \times 10^1 \times 10^{13} \]Combine the exponents: \[1.92 \times 10^{14}\]

Key Concepts

Significant Figures

In scientific notation, significant figures are the digits that carry meaning contributing to the precision of the number. These include all non-zero digits, any zeros between them, and any trailing zeros in the decimal part. In our example, we multiplied 6.4 by 3.

Both 6.4 and 3 have significant figures. The result of this multiplication, 19.2, also maintains significant figures. Each step must respect significant figures to ensure precision and accuracy.

When multiplying, add significant figures of the results to reflect the precision of your general calculation.

Exponents Addition

When dealing with numbers in scientific notation, a key step is to add the exponents. Scientific notation is a way of expressing very large or very small numbers by combining a coefficient (significant figures) with a power of ten.

In our exercise, the two numbers given were \(6.4 \times 10^5\) and \(3 \times 10^8\). Upon multiplying the significant figures (6.4 and 3), we get 19.2. Then we add the exponents:
5 for the first term and 8 for the second term, giving us:
\(5 + 8 = 13\)
This means that \(\6.4 \times 10^5\) multiplied by \(\3 \times 10^8\) equals \(19.2 \times 10^{13}\). Make sure to properly handle exponents addition to maintain accurate calculations when working with scientific notation.

Normalizing Scientific Notation

The final step in our calculation is to normalize the scientific notation. Normalizing means adjusting the number so it adheres to proper scientific notation format, which is a number between 1 and 10 multiplied by a power of ten.

Following the example, we initially have \(19.2 \times 10^{13}\). To normalize it, recognize that 19.2 can be written as \(1.92 \times 10^1\). Therefore, our expression becomes \(1.92 \times 10^1 \times 10^{13}\).
When you combine the exponents, adding 1 from \[10^1\] to 13 from \[10^{13}\], you get:
\(1 + 13 = 14\)Thus, the properly normalized form is \(1.92 \times 10^{14}\). Normalizing ensures that the number is in the correct format, making it easier to read and compare with other scientific notations.