Question

Perform the following operation. Express the answer in scientific notation and with the correct number of significant figures. 0.002115 \(\mathrm{m} \times 0.0000405 \mathrm{m}\)

Short answer

8.57 \times 10^{-8}

Step-by-step solution

Understand the problem

The task is to multiply two numbers, 0.002115 m and 0.0000405 m, and express the result in scientific notation with the correct number of significant figures.

Determine the significant figures

Identify the significant figures in each number. For 0.002115, there are 4 significant figures. For 0.0000405, there are 3 significant figures.

Perform the multiplication

Multiply the two numbers: \[ 0.002115 \times 0.0000405 = 8.56575 \times 10^{-8} \]

Apply significant figures

The result must be expressed with the same number of significant figures as the factor with the fewest significant figures, which is 3. Round 8.56575 to 3 significant figures, resulting in 8.57.

Convert to scientific notation

Express the correctly rounded result in scientific notation: \[ 8.57 \times 10^{-8} \]

Key concepts

significant figures

Significant figures (often abbreviated as 'sig figs') are the digits in a number that carry meaningful information about its precision. They include all non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number. Understanding significant figures is crucial for precision in calculations. In our problem, 0.002115 has 4 significant figures because the leading zeros are not counted. On the other hand, 0.0000405 has 3 significant figures. When performing multiplication or division, the result should be rounded to the same number of significant figures as the factor with the fewest significant figures. This ensures the precision of your result is appropriate to the precision of the original data.

multiplication of decimals

Multiplying decimal numbers involves a few simple steps:

• First, ignore the decimal points and multiply the numbers as if they were whole numbers.
• Next, count the total number of decimal places in both original numbers.
• Finally, place the decimal point in the product so that it has the same total number of decimal places.

In our example, 0.002115 (which has 4 decimal places) is multiplied by 0.0000405 (which has 7 decimal places), resulting in a raw product of 8.56575. This product has a combined total of 11 decimal places, which matches the combined decimal places of the numbers being multiplied. Make sure you keep track of these decimal places to ensure your result is accurate.

scientific notation conversion

Scientific notation is a way to express very large or very small numbers concisely. It’s written as the product of a number between 1 and 10, and a power of 10. For example, 8,570,000 can be written as 8.57 × 10⁶. Converting a number to scientific notation involves a few steps:

• Identify the most significant digit and place the decimal point immediately after it.
• Count how many places you moved the decimal point—this will be your exponent on 10.

In our problem, after rounding the product 8.56575 to 3 significant figures, we get 8.57. Then, we write 8.57 × 10^-8 to convert it into scientific notation. This compact form makes it easier to read and work with very small or very large numbers, such as in scientific experiments and engineering.