What is the product of 0.8102 \(\mathrm{m}\) and 3.44 \(\mathrm{m} ?\)

When multiplying decimal numbers, proper placement of the decimal point is crucial for getting the correct answer. Unlike whole numbers, decimals indicate fractions of a whole, which are often encountered in practical situations like measurements. The process involves a two-step approach: multiply the numbers as if they are whole numbers, and then adjust for the decimal points.

For instance, to find the product of 0.8102 and 3.44, first multiply 8102 by 344, ignoring the decimal points. You get 2785488. Then, count the total number of decimal places in the factors—in this case, 0.8102 has four decimal places, and 3.44 has two, making a total of four (since the decimal places in a whole number like 344 count as zero). Place the decimal in the result to give it four places from the right, resulting in 278.5488. It's key to keep track of the number of decimal places to ensure accuracy in the final result.

Scientific notation is a method of expressing numbers that are too large or too small in a compact and convenient format. It uses powers of ten and consists of two parts: a decimal part (a number between 1 and 10) and an exponential part, which indicates the power of ten. For example, the number 278.5488 can be written in scientific notation as \(2.785488 \times 10^2\).

This notation makes it easy to work with very large or very small numbers, especially in fields like science and engineering. When multiplying numbers in scientific notation, you multiply the decimal parts and then add the exponents of ten. It's a handy technique for managing unwieldy figures without losing precision.

After calculating a mathematical operation like multiplication, it's important to present the answer with the appropriate units, especially when dealing with real-world quantities. In our example, since we are multiplying two lengths measured in meters, the resulting unit is the square meter \(\mathrm{m}^2\), which represents an area.

For conversions between different units of measurement, one must use conversion factors that relate one unit to another. For instance, to convert square meters to square feet, you can multiply by the conversion factor which is approximately 10.7639 square feet per square meter. Understanding units is essential for interpreting results properly and ensuring they make sense in their given context.