write this in scientific notation with one digit before the decimal place 0.005030 , 0.004502
express the answer with correct number of significant figures assume that number represent measurements
1) (34.7 X 4.26x 10-3 X 55.97) / ( 0.020 X 16.3x 104 )
2) (4.5 X 350 X 3.832) / ( 5 x 47.3 )
The correct answer and explanation is :
To express the numbers 0.005030 and 0.004502 in scientific notation with one digit before the decimal place:
- 0.005030: Move the decimal point three places to the right to get 5.030, which corresponds to multiplying by 10³. Therefore, 0.005030 = 5.030 × 10⁻³. This number has four significant figures: 5, 0, 3, and 0.
- 0.004502: Similarly, move the decimal point three places to the right to get 4.502, corresponding to multiplying by 10³. Thus, 0.004502 = 4.502 × 10⁻³. This number also has four significant figures: 4, 5, 0, and 2.
Calculations:
- Numerator: 34.7 × 4.26 × 10⁻³ × 55.97 = 8,292.78018 × 10⁻³ = 8.29278018
- Denominator: 0.020 × 16.3 × 10⁴ = 32600 × 10⁻² = 326.0 Therefore, the result is approximately 0.0025379132944785276. Considering significant figures:
- The numerator has three significant figures (34.7), three significant figures (4.26 × 10⁻³), and four significant figures (55.97). The limiting factor is three significant figures.
- The denominator has two significant figures (0.020) and three significant figures (16.3 × 10⁴), so the limiting factor is two significant figures. The result should be expressed with two significant figures: 0.0025.
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- Numerator: 4.5 × 350 × 3.832 = 6,037.8
- Denominator: 5 × 47.3 = 236.5 Therefore, the result is approximately 25.51966173361522. Considering significant figures:
- The numerator has two significant figures (4.5), two significant figures (350), and four significant figures (3.832). The limiting factor is two significant figures.
- The denominator has one significant figure (5) and three significant figures (47.3), so the limiting factor is one significant figure. The result should be expressed with one significant figure: 30.
Explanation:
Scientific Notation: This is a method of writing numbers that accommodates values too large or small to be conveniently written in standard decimal form. It expresses numbers as a product of two factors: a coefficient (a number between 1 and 10) and a power of ten. For example, 0.005030 can be written as 5.030 × 10⁻³.
Significant Figures: These are the digits in a number that carry meaningful information about its precision. The rules for determining significant figures include:
- All non-zero digits are significant.
- Any zeros between significant digits are significant.
- Leading zeros (zeros before non-zero digits) are not significant.
- Trailing zeros in a number with a decimal point are significant.
In calculations:
- For multiplication and division, the result should have the same number of significant figures as the factor with the fewest significant figures.
- For addition and subtraction, the result should be rounded to the least precise decimal place.
Application to Calculations:
- In the first calculation, the numerator’s factors have 3, 3, and 4 significant figures, respectively, so the numerator is limited to 3 significant figures. The denominator’s factors have 2 and 3 significant figures, so the denominator is limited to 2 significant figures. Therefore, the final result should be rounded to 2 significant figures.
- In the second calculation, the numerator’s factors have 2, 2, and 4 significant figures, respectively, so the numerator is limited to 2 significant figures. The denominator’s factors have 1 and 3 significant figures, so the denominator is limited to 1 significant figure. Therefore, the final result should be rounded to 1 significant figure.
Understanding and applying the rules of significant figures and scientific notation ensures precision and clarity in scientific measurements and calculations.