Question

What is the sum of \(5.786 \cdot 10^{3} \mathrm{~m}\) and \(3.19 \cdot 10^{4} \mathrm{~m} ?\) a) \(6.02 \cdot 10^{23} \mathrm{~m}\) c) \(8.976 \cdot 10^{3} \mathrm{~m}\) b) \(3.77 \cdot 10^{4} \mathrm{~m}\) d) \(8.98-10^{3} \mathrm{~m}\)

Short answer

Answer: (b) \(3.77 \cdot 10^{4} \mathrm{~m}\).

Step-by-step solution

Convert to common base of powers of 10

To add these two numbers, we first need to convert them in such a way that they both have the same base of powers of 10. We want to change both numbers to format \(a \cdot 10^{b} \mathrm{~m}\), and then simply add them together.

Adjust the smaller number's exponent

We have \(5.786 \cdot 10^{3}\) and \(3.19 \cdot 10^{4}\). Since the second number has a greater exponent, we will adjust the first number's exponent. Divide the first number by 10 and increase its exponent by 1: \(\frac{5.786 \cdot 10^{3} \mathrm{~m}}{10}=0.5786 \cdot 10^{4} \mathrm{~m}\)

Add the two numbers

Now that we have the same powers of 10 for both numbers, we can add them together: \(0.5786 \cdot 10^{4} \mathrm{~m} + 3.19 \cdot 10^{4} \mathrm{~m} = (0.5786 + 3.19) \cdot 10^{4} \mathrm{~m} =3.7686 \cdot 10^{4} \mathrm{~m}\)

Convert to the answer format

Round the final result to two decimal places to match the format of the answer choices: \(3.7686 \cdot 10^{4} \mathrm{~m} \approx 3.77 \cdot 10^{4} \mathrm{~m}\) The correct answer is (b) \(3.77 \cdot 10^{4} \mathrm{~m}\).

Key concepts

Base Powers of 10

When working with large or small numbers in science and mathematics, we use scientific notation to make them more manageable. Scientific notation expresses numbers as a product of a number between 1 and 10, and a power of 10. This notation is particularly useful because it simplifies calculations, such as addition or multiplication, especially when dealing with extremely large or small values.

For example, the number \( 2,000 \) would be written in scientific notation as \( 2 \times 10^3 \). Here, \( 2 \) is the coefficient and \( 10^3 \) indicates the number is a base power of 10, with the exponent 3 revealing how many places the decimal has moved to the right. A base power of 10 always follows this format, where 10 is raised to an exponent to represent the magnitude of the number.

Exponent Adjustment

Exponent adjustment is a critical step when adding or subtracting numbers in scientific notation, as it ensures that both numbers have the same power of 10. This is necessary because it aligns the place values, making the addition straightforward.

To adjust the exponents, you may need to shift the decimal point of the number with the smaller exponent. For every shift to the right, you increase the exponent by 1; conversely, for each shift to the left, you decrease the exponent by 1. After adjusting, you can directly add or subtract the coefficients.

Example of Adjusting Exponents:

Suppose you have \( 4.5 \times 10^6 \) and \( 3 \times 10^5 \). You'd adjust the second number by shifting the decimal one place to the right, creating \( 0.3 \times 10^6 \) to match the exponent of the first number. Now you can add the coefficients \( 4.5 + 0.3 \) to find the sum.

Significant Figures

Significant figures are a way of expressing precision in a number. They indicate the meaningful digits which are important in a given measurement or calculation. The rules of significant figures dictate how numbers should be rounded during various operations, ensuring that the precision of the results reflects the precision of the inputs.

When adding or subtracting numbers, the result should be rounded to the least precise measurement in terms of decimal places. In the original exercise, when we add \( 0.5786 \) and \( 3.19 \)—after adjusting exponents—the result is \( 3.7686 \). However, the least precise number, \( 3.19 \) has two decimal places. Therefore, the final answer is rounded to two decimal places as well, resulting in \( 3.77 \), to maintain consistency with the significant figures.
Keeping track of significant figures ensures that the final answer is as accurate as possible given the precision of the numbers used in the calculation.