The numerical value [tex]$\left(5.6 \times 10^4\right) \div\left(7.89 \times 10^2\right)$[/tex] is equal to, with the proper number of significant figures:
A. 70.98
B. 71
C. [tex]$7.098 \times 10^1$[/tex]
D. 71.0
E. 70.976
Answer (1)
Calculate the numerical value: 7.89 × 1 0 2 5.6 × 1 0 4 ≈ 70.9759 .
Determine the number of significant figures: 5.6 has two, 7.89 has three.
Round the result to two significant figures: 71.
The final answer is 71 .
Explanation
Understanding the Problem We are asked to evaluate the expression 7.89 × 1 0 2 5.6 × 1 0 4 and express the result with the correct number of significant figures.
Calculating the Value First, let's calculate the numerical value of the expression: 7.89 × 1 0 2 5.6 × 1 0 4 = 789 56000 ≈ 70.9759
Considering Significant Figures Now, we need to consider significant figures. The number 5.6 has two significant figures, and the number 7.89 has three significant figures. When dividing, the result should have the same number of significant figures as the number with the fewest significant figures used in the calculation. In this case, 5.6 has the fewest significant figures (two).
Rounding to Significant Figures Therefore, we need to round the result to two significant figures. Rounding 70.9759 to two significant figures gives us 71.
Final Answer Thus, the numerical value of the expression 7.89 × 1 0 2 5.6 × 1 0 4 with the proper number of significant figures is 71.
Examples
In scientific calculations, it's crucial to maintain the correct number of significant figures to reflect the precision of measurements. For instance, if you're calculating the density of a material using mass and volume measurements, the density should be reported with the same number of significant figures as the least precise measurement. This ensures that the calculated result doesn't imply a higher level of accuracy than the original measurements allow. This principle is vital in fields like chemistry, physics, and engineering, where accurate data representation is essential for reliable results and decision-making.
