Question

Indicate the correct number of significant figures for each answer, given that all values are measurements: (a) \(\left(6.350 \times 10^{-8}\right) \times(0.0080)\) (b) \(\left(5.30 \times 10^{2}\right)+\left(22.1 \times 10^{2}\right)\) (c) \(\left(5.830 \times 10^{2}\right)+\left(22.100 \times 10^{2}\right)\) (d) \(\frac{100.0 \times 0.1500}{58.443}\) (e) \(\frac{100.0 \times 0.15}{58.4}\)

Short answer

The correct number of significant figures for each answer is: (a) 2 significant figures (b) 1 decimal place (c) 2 decimal places (d) 4 significant figures (e) 2 significant figures

Step-by-step solution

Identify the significant figures in the values

The given values are: - \(6.350 \times 10^{-8}\) has 4 significant figures - \(0.0080\) has 2 significant figures (b)

Identify the decimal places in the values

The given values are: - \(5.30 \times 10^{2}\) has 1 decimal place - \(22.1 \times 10^{2}\) has 1 decimal place (c)

Identify the decimal places in the values

The given values are: - \(5.830 \times 10^2\) has 2 decimal places - \(22.100 \times 10^2\) has 3 decimal places (d)

Identify the significant figures in the values

The given values are: - \(100.0\) has 4 significant figures - \(0.1500\) has 4 significant figures - \(58.443\) has 5 significant figures (e)

Identify the significant figures in the values

The given values are: - \(100.0\) has 4 significant figures - \(0.15\) has 2 significant figures - \(58.4\) has 3 significant figures Now, we apply the rules for each operation. (a)

Multiply the values and determine the significant figures

\((6.350 \times 10^{-8}) \times (0.0080)\) Since the least number of significant figures is 2, the result will have 2 significant figures. (b)

Add the values and determine the decimal places

\((5.30 \times 10^{2})+(\left(22.1 \times 10^{2}\right)\) As both values have 1 decimal place, the result will have 1 decimal place. (c)

Add the values and determine the decimal places

\((5.830 \times 10^{2})+\left(22.100 \times 10^{2}\right)\) Since the least number of decimal places is 2, the result will have 2 decimal places. (d)

Perform the division and determine the significant figures

\(\frac{100.0 \times 0.1500}{58.443}\) Since the least number of significant figures is 4, the result will have 4 significant figures. (e)

Perform the division and determine the significant figures

\(\frac{100.0 \times 0.15}{58.4}\) As the least number of significant figures is 2, the result will have 2 significant figures.

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used by scientists, engineers, and mathematicians to make calculations easier and to neatly capture the precision of measurements.

In scientific notation, a number is written as the product of two factors: a decimal greater than or equal to 1 and less than 10, and a power of 10. For example, the number 650,000 is written as \(6.5 \times 10^{5}\) in scientific notation, where the exponent 5 indicates that the decimal point has been moved 5 places to the right.

Using scientific notation simplifies operations involving very large or very small numbers and helps in identifying significant figures. In the exercise example \(6.350 \times 10^{-8}\), scientific notation clearly shows there are four significant figures in the measurement.

Significant Figures Rules

The rules for significant figures are important for accurately reporting measurements and ensuring consistent precision in calculations. Significant figures include all the digits that are known, plus a last digit that is estimated. Here are some key rules:

• Non-zero digits are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros, which precede all non-zero digits, are not significant.
• Trailing zeros in a decimal number are significant.
• Exact numbers, such as the number of objects or people, have an infinite number of significant figures.

In the exercise, for instance, \(0.0080\) has two significant figures because the zeros after the decimal and before 8 are not significant, but the trailing zero is.

Mathematical Operations with Significant Figures

When performing mathematical operations, significant figures play an important role in determining the precision of the result. Here's how to handle significant figures in different operations:

• In addition and subtraction, the result should have the same number of decimal places as the number with the fewest decimal places.
• In multiplication and division, the result should have the same number of significant figures as the number with the fewest significant figures.

For example, in the division exercise \(100.0 \times 0.1500 \div 58.443\), the number of significant figures in the result is based on the value with the fewest significant figures among the operands, which in this case, are the numbers being divided and multiplied. The result should thus have four significant figures to reflect the precision of the least precise measurement involved.