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Significant figures can be tricky, but they boil down to consistency in precision. In addition and subtraction, focus on the number of decimal places rather than the total number of significant figures. The rule is simple: the result should match the number with the fewest decimal places from your input figures.
Let's look at this with an example from our exercise: you're asked to solve the equation \(x = 17.2 + 65.18 - 2.4\). In this case, both 17.2 and 2.4 have just 1 decimal place, while 65.18 has 2 decimal places. Thus, your result should end with 1 decimal place. After performing the calculation, you get 79.98. By rounding it to 1 decimal place, the final answer is 80.0.

• Always line up your numbers by their decimal points when adding or subtracting.
• Perform the calculation to full precision and then round.
• Check which number has the fewest decimal places before rounding your answer.

This ensures your answer is not falsely precise, staying true to the certainty of the original numbers.

For division, the focus shifts to the total number of significant figures. Your result should share the same number of significant figures as the least precise figure in your calculation.
Take the example \(x = \frac{13.0217}{17.10}\). In this equation, 13.0217 boasts 6 significant figures, while 17.10 has 4 significant figures. Therefore, the answer must be calculated to 4 significant figures because it's the smallest count among the numbers used.
Perform the division to get 0.761801. Rounding this to 4 significant figures results in 0.7618.

• Identify the number with the lowest count of significant figures among your values.
• Calculate the result to full precision initially.
• Round your final answer to match the smallest significant figure count.

Remember, this method helps you respect the accuracy limits of the original measurements.

When multiplying, the rule for significant figures is quite similar to that for division: your result should reflect the number with the fewest significant figures from the numbers in your calculation.
For instance, consider the problem \(x = (0.0061020)(2.0092)(1200.00)\). Here, 0.0061020 and 2.0092 both have 5 significant figures, while 1200.00 stands out with 6 significant figures. Thus, the result must have 5 significant figures to meet the criteria.
Multiplying these numbers yields the raw result of 14.69400784. When rounded to 5 significant figures, the result is 14.694.

• Check for the smallest significant figures among your numbers before proceeding.
• Compute fully, then round your result to comply with the least precise figure count.
• This ensures no overconfidence in the exactness of your result.

Precision in multiplication is crucial, ensuring your outcome is as accurate as the least precise element in your equation.

Combining all these operations in complex expressions can seem daunting, but it becomes manageable by dealing with each component separately. Use the respective rules for addition, subtraction, multiplication, and division on each part of the expression.
Consider the expression \(x = 0.0034 + \frac{\sqrt{(0.0034)^2 + 4(1.000)(6.3 \times 10^{-4})}}{2(1.000)}\).
Initially, calculate inside the square root: \((0.0034)^2 + 4(1.000)(6.3 \times 10^{-4})\). Once calculated, you have 0.00253156. After finding the square root with respect to significant figures, consider it as having two significant figures yielding 0.0503147. Divide by 2 to keep 0.02515735, rounding it to 0.025.
In the final step, add 0.0034 maintaining precision to two significant figures, resulting in 0.0284.

• Break down the expression, solving from the innermost parts outward.
• Apply the relevant rule for each mathematical operation.
• Consistently check for the appropriate number of significant figures.

This approach ensures accuracy while solving intricate numerical expressions, adhering to the different rules for each mathematical operation involved.