Express the number 0.000216 in scientific notation (2.16 x 10^-4). Express 166,000 in exponential (scientific) notation (1.66 x 10^5). Express the number 8.64 x 10^-3 in common decimal form (0.00864). Express the number 1.16 x 10^4 in common decimal form (11600). How many significant figures are there in the number 0.045607007 (7)?
The Correct Answer and Explanation is:
ChatGPT said:
Correct Answers:
- 0.000216 in scientific notation:
👉 2.16 × 10⁻⁴ - 166,000 in scientific notation:
👉 1.66 × 10⁵ - 8.64 × 10⁻³ in decimal form:
👉 0.00864 - 1.16 × 10⁴ in decimal form:
👉 11,600 - Significant figures in 0.045607007:
👉 7 significant figures
Explanation
Scientific notation is a way of expressing very large or very small numbers using powers of ten. It helps in simplifying calculations and presenting data clearly. A number in scientific notation is typically written as:a×10na \times 10^na×10n
where 1 ≤ a < 10, and n is an integer.
- For 0.000216, you move the decimal point 4 places to the right to get 2.16, so the exponent is –4. Hence, it’s written as 2.16 × 10⁻⁴.
- For 166,000, move the decimal point 5 places to the left to get 1.66. The exponent is +5, so it becomes 1.66 × 10⁵.
Converting scientific notation to decimal involves reversing the exponent:
- 8.64 × 10⁻³: Move the decimal 3 places left, getting 0.00864.
- 1.16 × 10⁴: Move the decimal 4 places right, resulting in 11,600.
Now, regarding significant figures, these are the digits in a number that carry meaningful information about its precision. The rules are:
- All non-zero digits are significant.
- Any zeros between non-zero digits are significant.
- Leading zeros (zeros before the first non-zero digit) are not significant.
- Trailing zeros after a decimal point are significant.
In 0.045607007, we identify:
- Ignore the leading zeros: ‘0.0’.
- Remaining digits: 4, 5, 6, 0, 7, 0, 0, 7.
- All these are either non-zero or meaningful zeros between or after significant digits.
So, the number has 7 significant figures: 4, 5, 6, 0, 7, 0, 0, 7 → 7 meaningful digits.
