Convert 0.0342 to scientific notation by adding the prefix and exponent values in the provided boxes..
[Fill in the blank]
x10^[Fill in the blank]

Question

Convert 0.0342 to scientific notation by adding the prefix and exponent values in the provided boxes..
[Fill in the blank]
 x10^[Fill in the blank]

BlackTom AI · Accepted Answer

Scientific notation requires a coefficient (also called the mantissa) between 1 and 10 and an integer power of ten that records how many places the decimal point was shifted.

1. Relevant concepts
- Scientific (normalized) notation: number = (mantissa) × 10^(exponent), with 1 ≤ mantissa < 10.
- Moving the decimal point to the right by n places multiplies the number by 10^n; moving it left by n places divides by 10^n (equivalently multiplies by 10^(-n)).

2. Step-by-step reasoning
Start with 0.0342. We need a mantissa between 1 and 10, so move the decimal point right until the first nonzero digit is immediately left of the decimal: moving it two places right gives 3.42. Because we moved the decimal right by 2 places, we multiplied the original number by 10^2; to keep the equality, we must multiply 3.42 by 10^(-2).

3. Intermediate calculations
0.0342 × 10^2 = 3.42
Therefore 0.0342 = 3.42 × 10^(-2).
Verify: 3.42 × 10^(-2) = 3.42 × 0.01 = 0.0342, which matches the original number.

4. Why this is correct
The mantissa 3.42 lies in the required range (1 ≤ mantissa < 10). The exponent −2 correctly undoes the two-place right shift of the decimal, restoring the original magnitude. The arithmetic verification above confirms equality.

5. Why other possibilities are incorrect
- Using exponent −1 (e.g., 0.342 × 10^(−1) or 3.42 × 10^(−1)) would produce 0.0342? Check: 3.42 × 10^(−1) = 0.342, not 0.0342 — incorrect.
- Using exponent −3 (e.g., 34.2 × 10^(−3)) could equal 0.0342 numerically (34.2×10^−3 = 0.0342), but 34.2 is not an allowed mantissa because it’s ≥10; normalized scientific notation requires 1 ≤ mantissa < 10.
- Using a positive exponent would give a much larger number, not 0.0342.

6. Conclusion
The correct normalized scientific notation is 3.42 × 10^(−2). This matches the provided answer: mantissa = 3.42 and exponent = -2.

Convert 0.0342 to scientific notation by adding the prefix and exponent values in the provided boxes.. [Fill in the blank] x10^[Fill in the blank] 多项填空题

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