) Express 9876000 in scientific notation: B) Express 0000235 in scientific notation: C) Express 762000 in scientific notation: D) Express
as a decimal: E) Express
as a decimal: F) Express
as a decimal: 3. Multiply the following: 4 point
The Correct Answer and Explanation is:
Solutions and Explanations
Here are the solutions to the problems presented on the worksheet. Each answer is followed by an explanation of the underlying principles of scientific notation.
A) Express 9876000 in scientific notation:
- Answer: 9.876 x 10⁶
B) Express .0000235 in scientific notation: - Answer: 2.35 x 10⁻⁵
C) Express 762000 in scientific notation: - Answer: 7.62 x 10⁵
Explanation for A, B, C: To write a number in scientific notation, the decimal point is moved to create a new number between 1 and 10. The number of places the decimal point was moved determines the exponent on the 10.
- In 9,876,000, the decimal is moved 6 places to the left to get 9.876. Because the original number was large, the exponent is positive: 9.876 x 10⁶.
- In 0.0000235, the decimal is moved 5 places to the right to get 2.35. Because the original number was small (less than 1), the exponent is negative: 2.35 x 10⁻⁵.
D) Express 7.3 x 10⁷ as a decimal:
- Answer: 73,000,000
E) Express 3.23 x 10⁻⁴ as a decimal: - Answer: 0.000323
F) Express 5.65 x 10⁻⁵ as a decimal: - Answer: 0.0000565
Explanation for D, E, F: To convert from scientific notation to a decimal, a positive exponent tells you to move the decimal point to the right, making the number larger. A negative exponent tells you to move it to the left, making it smaller.
- For 7.3 x 10⁷, the exponent is +7, so the decimal moves 7 places to the right: 73,000,000.
- For 3.23 x 10⁻⁴, the exponent is -4, so the decimal moves 4 places to the left: 0.000323.
3. Multiply the following:
- 5.1 x 10⁵ x 1.4 x 10² = 7.14 x 10⁷
- 1.2 x 10⁶ x 9.9 x 10⁻³ = 1.188 x 10⁴
Explanation: To multiply numbers in scientific notation, multiply the coefficients (the decimal parts) and add the exponents.
- (5.1 x 1.4) = 7.14. For the exponents: 10⁵ x 10² = 10⁵⁺² = 10⁷. The result is 7.14 x 10⁷.
- (1.2 x 9.9) = 11.88. For the exponents: 10⁶ x 10⁻³ = 10⁶⁺⁽⁻³⁾ = 10³. This gives 11.88 x 10³, which must be corrected to proper scientific notation. Move the decimal in 11.88 one place left to get 1.188, and add 1 to the exponent: 1.188 x 10⁴.
4. Divide:
- (9.6 x 10⁵) / (1.4 x 10³) = 6.86 x 10² (rounded)
- (3.9 x 10⁻⁴) / (1.3 x 10⁻⁹) = 3 x 10⁵
Explanation: To divide numbers in scientific notation, divide the coefficients and subtract the exponents (denominator from numerator).
- 9.6 ÷ 1.4 ≈ 6.86. For the exponents: 10⁵ / 10³ = 10⁵⁻³ = 10². The result is 6.86 x 10².
- 3.9 ÷ 1.3 = 3. For the exponents: 10⁻⁴ / 10⁻⁹ = 10⁻⁴⁻⁽⁻⁹⁾ = 10⁵. The result is 3 x 10⁵.
5. Which is the larger amount: 3.2 x 10⁴ or 5.8 x 10⁻⁷? How do you know?
- Answer: 3.2 x 10⁴ is the larger amount.
- Reason: When comparing numbers in scientific notation, the number with the larger exponent of 10 is always the larger value. Since 4 is greater than -7, 3.2 x 10⁴ (which is 32,000) is significantly larger than 5.8 x 10⁻⁷ (which is 0.00000058).
6. What is the rule about how many numbers to the left of the decimal in scientific notation?
- Answer: There must be exactly one non-zero digit to the left of the decimal point.
B. If you have 32.3 x 10⁶ how do you correct this?
- Answer: To correct this, you move the decimal one place to the left in the coefficient (32.3 becomes 3.23) and increase the exponent by one (6 becomes 7). The correct form is 3.23 x 10⁷.
- Explanation: The coefficient must be a number between 1 and 10. By making the coefficient 10 times smaller (dividing by 10), you must make the power of 10 ten times larger (by adding 1 to the exponent) to ensure the overall value remains the same.
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