Let's go through the questions shown in the image one by one.

1. Explain the relationship between a number in standard form and the sign of the exponent when the number is written in scientific notation.

• Answer:
  • In scientific notation, the exponent reflects how many times the base (10) is multiplied by itself.
  • A positive exponent indicates that the decimal point has been moved to the right (the number is greater than 1).
  • A negative exponent shows that the decimal point has been shifted to the left (the number is less than 1).

2. Open-ended: Write a number in standard form and then write the number in scientific notation, explaining each step that you used.

• Example:
  • Standard form: 12,300,000
  • Scientific notation: $1.23 \times 10^7$
  • Steps:
    1. Identify the first non-zero digit (1).
    2. Place the decimal point after the first digit to get $1.23$.
    3. Count the number of decimal places you moved the decimal point from the original position to reach the new location (7 places).
    4. Since the number is large (greater than 1), the exponent is positive.

3. Express $3.08 \times 10^{-4}$ in standard form.

• Answer: $3.08 \times 10^{-4} = 0.000308$ The exponent is negative, so we move the decimal point 4 places to the left.

4. Express $1.4 \times 10^2$ in standard form.

• Answer: $1.4 \times 10^2 = 140$ The exponent is positive, so we move the decimal point 2 places to the right.

5. Express $8.495 \times 10^5$ in standard form.

• Answer: $8.495 \times 10^5 = 849,500$ The exponent is positive, so we move the decimal point 5 places to the right.

6. Express 80,000,000 in scientific notation.

• Answer: $80,000,000 = 8.0 \times 10^7$ The decimal point is moved 7 places to the left.

7. Express 697,000 in scientific notation.

• Answer: $697,000 = 6.97 \times 10^5$ The decimal point is moved 5 places to the left.

8. Express 0.059 in scientific notation.

• Answer: $0.059 = 5.9 \times 10^{-2}$ The decimal point is moved 2 places to the right, so the exponent is negative.

9. The diameter of a spider's thread is 0.001 inch. Express in scientific notation.

• Answer: $0.001 = 1.0 \times 10^{-3}$ The decimal point is moved 3 places to the right.

10. Space: Refer to the table in Example 3 on page 187. To the nearest second, how long does it take light to travel from the Sun to Earth?

• Without the book reference, I can't provide an exact answer. However, the approximate time for light to travel from the Sun to Earth is 499 seconds or about 8 minutes and 19 seconds.

Would you like a detailed explanation on any of these? Here are some relative questions for more practice:

1. How would you express $6.02 \times 10^{23}$ in standard form?
2. Convert 0.00045 into scientific notation.
3. Explain how scientific notation helps in representing very large or very small numbers.
4. What is $5.67 \times 10^{-3}$ in standard form?
5. Express 0.00000032 in scientific notation.

Tip: When dealing with scientific notation, always check the direction of the decimal movement—left for negative exponents, right for positive exponents!