Question

Perform the following mathematical operations, and express each result to the correct number of significant figures. a. $\frac{0.102\times 0.0821\times 273}{1.01}$ b. $0.14\times 6.022\times {10}^{23}$ c. $4.0\times {10}^4\times 5.021\times {10}^{-3}\times 7.34993\times {10}^2$ d. $\frac{2.00\times {10}^6}{3.00\times {10}^{-7}}$

Short answer

The answers are: a. 2.18 b. $8.4\times {10}^{23}$ c. $1.5\times {10}^6$ d. $6.67\times {10}^9$

Step-by-step solution

a. 0.102 x 0.0821 x 273 / 1.01

To multiply and divide numbers with significant figures: 1. Count the number of significant figures in each of the numbers, and note the one with the least number of significant figures. 2. Perform the requested operations (multiplication and division). 3. Round the result to the least number of significant figures noted earlier. Applying the above rules to the given problem: 1. Counting the significant figures of each number: 0.102 (3 significant figures), 0.0821 (4 significant figures), 273 (3 significant figures), and 1.01 (3 significant figures). 2. Perform the operations: ${\displaystyle \frac{0.102\times 0.0821\times 273}{1.01}}=2.17889908616187969$ 3. 0.102 has the least number of significant figures (3), so round the result to 3 significant figures: 2.18

b. 0.14 x 6.022 x 10^23

To multiply numbers with significant figures: 1. Count the significant figures in each number, and note the one with the least number of significant figures. 2. Perform the requested operation (multiplication). 3. Round the result to the least number of significant figures noted earlier. Applying the above rules to the given problem: 1. Counting the significant figures of each number: 0.14 (2 significant figures), and 6.022 x 10^23 (4 significant figures). 2. Perform the operation: $0.14\times 6.022\times {10}^{23}=8.4308\times {10}^{23}$ 3. 0.14 has the least number of significant figures (2), so round the result to 2 significant figures: $8.4\times {10}^{23}$

c. 4.0 x 10^4 x 5.021 x 10^-3 x 7.34993 x 10^2

To multiply numbers with significant figures: 1. Count the significant figures in each number, and note the one with the least number of significant figures. 2. Perform the requested operation (multiplication). 3. Round the result to the least number of significant figures noted earlier. Applying the above rules to the given problem: 1. Counting the significant figures of each number: 4.0 x 10^4 (2 significant figures), 5.021 x 10^-3 (4 significant figures), and 7.34993 x 10^2 (6 significant figures). 2. Perform the operation: $4.0\times {10}^4\times 5.021\times {10}^{-3}\times 7.34993\times {10}^2=1470209.96$ 3. 4.0 x 10^4 has the least number of significant figures (2), so round the result to 2 significant figures: $1.5\times {10}^6$

d. 2.00 x 10^6 / 3.00 x 10^-7

To divide numbers with significant figures: 1. Count the number of significant figures in each of the numbers, and note the one with the least number of significant figures. 2. Perform the requested operation (division). 3. Round the result to the least number of significant figures noted earlier. Applying the above rules to the given problem: 1. Counting the significant figures of each number: 2.00 x 10^6 (3 significant figures) and 3.00 x 10^-7 (3 significant figures). 2. Perform the operation: ${\displaystyle \frac{2.00\times {10}^6}{3.00\times {10}^{-7}}}=6666666666.66667$ 3. Both numbers have the same number of significant figures (3), so round the result to 3 significant figures: $6.67\times {10}^9$

Key concepts

Understanding Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are fundamental when working with numbers. In this exercise, we focused on multiplication and division, performed on numbers with varying significant figures.

• Multiplication involves combining the units of the quantities, resulting in a product, while division separates the quantity into equal parts, resulting in a quotient.
• When performing these operations with numbers, it's crucial to consider the rules of significant figures.
• For instance, in multiplication and division, the number of significant figures in the final result is determined by the number with the least significant figures in the calculation.

This ensures precision and maintains the integrity of the measurements involved, as seen in the examples provided where each operation's result was rounded appropriately.

Exploring Scientific Notation

Scientific notation is a method of expressing very large or very small numbers, making them easier to work with. It employs a base number and an exponent, usually in the form of a digit times ten raised to a power, like $6.022\times {10}^{23}$.

• This format is especially useful in scientific calculations to simplify multiplication and division, as the exponents are simply added or subtracted.
• For example, when multiplying $4.0\times {10}^4$ by $5.021\times {10}^{-3}$, even though these numbers are very different in size, they can easily be calculated using their exponent forms.
• In scientific notation, each part of the number holds significance, and the digits displayed in the base number are crucial for determining the significant figures.

This streamlined approach is particularly beneficial in scientific fields and helps maintain accuracy throughout complex calculations.

Mastering Rounding Rules

The art of rounding applies fundamental rules to decide how many digits to keep in a number. These rules ensure that the numbers are expressed with the correct amount of precision.

• In general, when a number ends in 5 or greater, you round up. If it's less than 5, you round down.
• For calculations involving significant figures, the quantity of significant figures from the original numbers affects the rounding.
• As demonstrated, when the number $2.17889908616187969$ was rounded, it was expressed as 2.18 to match the three significant figures from the number 0.102.

These rounding practices prevent the inflation of precision that wasn't present in the original numbers and ensure the calculations remain reliable.

Precision in Calculations

Precision in calculations indicates the degree to which repeated measurements under unchanged conditions show the same results. When dealing with significant figures, precision reflects the consistency and reliability of numerical data.

• Each measurement or operation in calculations can affect the overall precision.
• In these exercises, determining which number has the least significant figures guides how precise the reported result should be.
• This is vital in scientific disciplines, where over-precision can mislead the data interpretation, making skilled usage of significant figures crucial.

Through these principles, more accurate and dependable results are achieved, ensuring findings reflected in studies and experiments are correctly interpreted.