Scientific Notation

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Scientific Notation · Grade 7 · Worksheet 3

1. Mason is studying the world's oceans for a science project. He learns that the Pacific Ocean contains approximately 6.6 × 10^8 cubic kilometers of water, while the Atlantic Ocean contains about 3.1 × 10^8 cubic kilometers of water. What is the total volume of water in both oceans combined? Express your answer in proper scientific notation. Answer: 9.7 × 10^8 Solution: Write down the volumes in scientific notation. Pacific Ocean: 6.6 × 10^8 cubic km Atlantic Ocean: 3.1 × 10^8 cubic km Since both numbers have the same exponent (10^8), we can add the coefficients directly.
   Full step-by-step solution

   Step 1: Write down the volumes in scientific notation. Pacific Ocean: 6.6 × 10^8 cubic km Atlantic Ocean: 3.1 × 10^8 cubic km Step 2: Since both numbers have the same exponent (10^8), we can add the coefficients directly. 6.6 + 3.1 = 9.7 Step 3: Keep the same power of 10. 9.7 × 10^8 Step 4: Check if the answer is in proper scientific notation. 9.7 is between 1 and 10, so 9.7 × 10^8 is already in proper form. The total volume of water in both oceans combined is 9.7 × 10^8 cubic kilometers.

2. Mere is helping her class learn about the distances to nearby stars. She reads that the star Alpha Centauri A is approximately 4.37 × 10^13 kilometers from Earth, and the star Sirius is approximately 8.61 × 10^13 kilometers from Earth. What is the combined distance from Earth to both stars? Express your answer in proper scientific notation. Answer: 1.298 × 10^14 Solution: Write the distances in scientific notation. Alpha Centauri A: 4.37 × 10^13 km Sirius: 8.61 × 10^13 km Since both numbers have the same exponent (10^13), add the coefficients directly.
   Full step-by-step solution

   Step 1: Write the distances in scientific notation. Alpha Centauri A: 4.37 × 10^13 km Sirius: 8.61 × 10^13 km Step 2: Since both numbers have the same exponent (10^13), add the coefficients directly. 4.37 + 8.61 = 12.98 Step 3: Write the sum with the same power of ten. 12.98 × 10^13 Step 4: Check if the answer is in proper scientific notation. The coefficient 12.98 is not between 1 and 10, so we must adjust. Move the decimal point one place to the left and increase the exponent by 1. 12.98 × 10^13 = 1.298 × 10^14 Step 5: Verify that 1.298 is between 1 and 10. It is, so the answer is in proper scientific notation. The combined distance is 1.298 × 10^14 kilometers.

3. A research lab is studying bacterial growth. They observe that a particular colony doubles in size every 3 hours. If the colony starts with 8,000 bacteria, how many bacteria will there be after 12 hours? Answer: 128000 Solution: We start with 8,000 bacteria. The colony doubles in size every 3 hours. We want the number of bacteria after 12 hours.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Understand the problem** We start with 8,000 bacteria. The colony doubles in size every 3 hours. We want the number of bacteria after 12 hours. --- **Step 2: Determine how many doubling periods occur in 12 hours** Doubling time = 3 hours. Total time = 12 hours. Number of doubling periods = 12 / 3 = 4. So the colony doubles 4 times. --- **Step 3: Apply the doubling** Start: 8,000 bacteria. After 1st doubling (3 hours): 8,000 × 2 = 16,000 After 2nd doubling (6 hours): 16,000 × 2 = 32,000 After 3rd doubling (9 hours): 32,000 × 2 = 64,000 After 4th doubling (12 hours): 64,000 × 2 = 128,000 --- **Step 4: Use the exponential growth formula to check** Initial amount = 8,000 Growth factor per period = 2 Number of periods = 4 Final amount = 8,000 × (2^4) = 8,000 × 16 = 128,000 --- **Step 5: Conclusion** After 12 hours, the colony will have 128,000 bacteria. **Final answer: 128000**

4. A research lab is studying bacterial growth. A particular colony starts with 8.5 × 10^6 bacteria and triples every hour. After 4 hours, how many bacteria are in the colony? Express your answer in scientific notation. Answer: 6.885 × 10^8 Solution: We start with \( 8.5 \times 10^6 \) bacteria. The colony triples every hour. We want the number after 4 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We start with \( 8.5 \times 10^6 \) bacteria. The colony triples every hour. We want the number after 4 hours. --- **Step 2: Set up the exponential growth formula** Exponential growth: Final amount = Initial amount × (growth factor)^(number of time periods) Here: Initial = \( 8.5 \times 10^6 \) Growth factor = 3 Time periods = 4 hours So: Final = \( 8.5 \times 10^6 \times 3^4 \) --- **Step 3: Calculate \( 3^4 \)** \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \) --- **Step 4: Multiply \( 8.5 \times 81 \)** \( 8.5 \times 80 = 680 \) \( 8.5 \times 1 = 8.5 \) Add: \( 680 + 8.5 = 688.5 \) So: \( 8.5 \times 81 = 688.5 \) --- **Step 5: Combine with \( 10^6 \)** We have \( 688.5 \times 10^6 \) --- **Step 6: Convert to proper scientific notation** \( 688.5 \times 10^6 = 6.885 \times 10^2 \times 10^6 \) \( = 6.885 \times 10^{8} \) --- **Final Answer:** \( 6.885 \times 10^8 \)

5. Hana is analyzing a scale model of the solar system in her science class. The model shows the distance from the Sun to Earth as 1.5 × 10^8 km, but the model's label has smudged and she can only read '150,000,000 km'. The model also shows the distance from the Sun to Jupiter as 7.8 × 10^8 km, but that label is also smudged. Help Hana by writing both distances in standard form and scientific notation, then answer: what is the sum of the two distances in scientific notation? Answer: 9.3 × 10^8 Solution: Write the distance from the Sun to Earth in scientific notation. The smudged label says 150,000,000 km.
   Full step-by-step solution

   Step 1: Write the distance from the Sun to Earth in scientific notation. The smudged label says 150,000,000 km. To write this in scientific notation, move the decimal point so that there is one non-zero digit to the left: 1.50000000. Count how many places you moved the decimal: 8 places to the left. So 150,000,000 = 1.5 × 10^8 km. This matches the model's 1.5 × 10^8 km. Step 2: Write the distance from the Sun to Jupiter in standard form. The model says 7.8 × 10^8 km. To convert to standard form, move the decimal point 8 places to the right: 7.8 → 780,000,000 km. Step 3: Add the two distances. Both are already in scientific notation with the same power of 10 (10^8). So add the coefficients: 1.5 + 7.8 = 9.3. Keep the same power of 10: 9.3 × 10^8 km. Step 4: Check that the answer is in proper scientific notation. 9.3 is between 1 and 10, so 9.3 × 10^8 is correct. The sum of the distances is 9.3 × 10^8 km.

6. Ava is researching the mass of the largest known star, UY Scuti. She reads that its estimated mass is approximately 1.7 × 10^31 kilograms. She also learns that the mass of Earth is about 5.97 × 10^24 kilograms. How many times more massive is UY Scuti than Earth? Express your answer in proper scientific notation (rounded to two decimal places). Answer: 2.85 × 10^6 Solution: Write the division problem: (1.7 × 10^31) ÷ (5.97 × 10^24). Divide the coefficients: 1.7 ÷ 5.97 ≈ 0.2848. Divide the powers of ten: 10^31 ÷ 10^24 = 10^(31-24) = 10^7.
   Full step-by-step solution

   Step 1: Write the division problem: (1.7 × 10^31) ÷ (5.97 × 10^24). Step 2: Divide the coefficients: 1.7 ÷ 5.97 ≈ 0.2848. Step 3: Divide the powers of ten: 10^31 ÷ 10^24 = 10^(31-24) = 10^7. Step 4: Combine: 0.2848 × 10^7. Step 5: Adjust to proper scientific notation (coefficient between 1 and 10): 0.2848 × 10^7 = 2.848 × 10^6. Step 6: Round to two decimal places: 2.85 × 10^6. The answer is 2.85 × 10^6.

7. Tane is researching the population of a rare bird species in New Zealand. He finds that there are approximately 9.7 × 10^5 birds in the North Island and 5.3 × 10^4 birds in the South Island. What is the total number of birds in both islands? Express your answer in proper scientific notation. Answer: 1.023 × 10^6 Solution: Write both numbers with the same exponent. The larger exponent is 5 (from 9.7 × 10^5). Convert 5.3 × 10^4 to have an exponent of 5: 5.3 × 10^4 = 0.53 × 10^5.
   Full step-by-step solution

   Step 1: Write both numbers with the same exponent. The larger exponent is 5 (from 9.7 × 10^5). Convert 5.3 × 10^4 to have an exponent of 5: 5.3 × 10^4 = 0.53 × 10^5. Step 2: Add the coefficients: 9.7 + 0.53 = 10.23. Step 3: Write the sum in proper scientific notation: 10.23 × 10^5 = 1.023 × 10^6. The answer is 1.023 × 10^6 birds.