Write an essay that compares and contrasts the process of adding or subtracting numbers written in scientific notation with the process of multiplying or dividing numbers written in scientific notation.

Question

Include the following;
How are the steps in each case similar? (2 points)
How are they different? (2 points)
Do you think adding, subtracting, multiplying, or dividing numbers written in scientific notation is easiest? Explain why you think it is the easiest? (2 points)
Examples using scientific notation to justify your reasoning. (2 points)

Answer 1

Comparing and Contrasting the Process of Adding or Subtracting Numbers Written in Scientific Notation with the Process of Multiplying or Dividing Numbers Written in Scientific Notation

Scientific notation is a method of writing numbers that are either very large or very small, using a combination of a coefficient and a power of 10. When it comes to performing operations like adding, subtracting, multiplying, and dividing numbers in scientific notation, there are both similarities and differences in the steps involved. In this essay, we will explore these similarities and differences and determine which operation is easiest to perform.

Starting with the similarities, both adding or subtracting and multiplying or dividing numbers in scientific notation require converting the numbers to have the same power of 10. This allows for a more straightforward manipulation of the coefficients. In both cases, it is essential to adjust the power of 10 to ensure that the numbers are compatible for the desired operation.

On the other hand, the differences between these two processes become evident when we dive into the specific steps involved. Adding or subtracting numbers in scientific notation requires aligning the decimals and performing the addition or subtraction operation between the coefficients while leaving the power of 10 unchanged. The result should also be in scientific notation, with the sum or difference adjusted accordingly.

In contrast, multiplying or dividing numbers in scientific notation involves multiplying or dividing the coefficients and adding or subtracting the powers of 10, respectively. The product or quotient is then expressed in scientific notation with the coefficient adjusted accordingly. The key distinction here is the manipulation of the powers of 10, which does not occur in the addition or subtraction process.

Determining which operation is easiest, adding, subtracting, multiplying, or dividing numbers written in scientific notation, is subjective and depends on one's mathematical proficiency. However, some argue that multiplying or dividing numbers in scientific notation may be easier due to the absence of the need to align decimals and perform operations with decimal fractions. Furthermore, multiplying and dividing are often seen as fundamentally simpler operations compared to adding or subtracting.

To exemplify this reasoning, consider the following examples:

1. Adding: 5 x 10^4 + 3 x 10^3
To add these numbers in scientific notation, we shift the decimal point in the second number by one place to the right to align the powers of 10, yielding 5 x 10^4 + 0.3 x 10^4. Next, we perform the addition operation on the coefficients, resulting in 5.3 x 10^4.

2. Multiplying: (6 x 10^2) x (2 x 10^3)
To multiply these numbers in scientific notation, we first multiply the coefficients, yielding 12 x 10^5. Next, we add the powers of 10, resulting in the final answer of 1.2 x 10^6.

As seen in these examples, multiplying the numbers in scientific notation involves a simpler process as compared to adding them. Additionally, the absence of aligning decimals and performing operations with fractional numbers makes multiplication or division in scientific notation appear more straightforward.

In conclusion, the process of adding or subtracting numbers written in scientific notation shares similarities with the process of multiplying or dividing them. However, the key differences lie in the manipulation of powers of 10 and the absence or presence of aligning decimals. While determining the easiest operation is subjective, some argue that multiplying or dividing numbers in scientific notation may be easier due to the simplified steps involved. Ultimately, the ease of performing these operations depends on an individual's mathematical skill and familiarity with scientific notation.