Scientific Notation

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Answer Key & Explanations

Scientific Notation · Grade 7 · Worksheet 1

1. Write 7,350,000 in scientific notation. Answer: 7.35 × 10⁶ Solution: Start with 7,350,000. The decimal point is at the end: 7,350,000. Move the decimal point left until only one non-zero digit remains to its left: 7.350000.
   Full step-by-step solution

   Step 1: Start with 7,350,000. The decimal point is at the end: 7,350,000. Step 2: Move the decimal point left until only one non-zero digit remains to its left: 7.350000. Step 3: Count the number of places moved: 6 places. Step 4: Write the number as 7.35 × 10⁶ (since 1 ≤ 7.35 < 10). The answer is 7.35 × 10⁶.

2. Lily is studying astronomy and learns that the distance from Earth to the Andromeda Galaxy is approximately 2.4 × 10^19 kilometers. If a space probe travels at a constant speed of 6 × 10^4 kilometers per hour, how many hours would it take to reach the Andromeda Galaxy? Express your answer in scientific notation. Answer: 4 × 10^14 Solution: Distance = 2.4 × 10^19 km Speed = 6 × 10^4 km/h Use the formula: Time = Distance ÷ Speed Time = (2.4 × 10^19) ÷ (6 × 10^4) Divide the coefficients: 2.4 ÷ 6 = 0.4 Subtract the exponents: 10^(19-4) = 10^15 Combine the results: 0.4 × 10^15 Convert to proper scientific notation: 0.4 × 10^15 = 4 ×…
   Full step-by-step solution

   Step 1: Write down the distance and speed Distance = 2.4 × 10^19 km Speed = 6 × 10^4 km/h Step 2: Use the formula: Time = Distance ÷ Speed Time = (2.4 × 10^19) ÷ (6 × 10^4) Step 3: Divide the coefficients: 2.4 ÷ 6 = 0.4 Step 4: Subtract the exponents: 10^(19-4) = 10^15 Step 5: Combine the results: 0.4 × 10^15 Step 6: Convert to proper scientific notation: 0.4 × 10^15 = 4 × 10^14 The answer is 4 × 10^14 hours.

3. A triangular prism is drawn with a right triangle base where the legs measure 6 cm and 8 cm, and the prism height is 15 cm. What is the volume of this prism in cubic centimeters? Answer: 360 Solution: Find the area of the triangular base. The base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 × 8 = (1/2) × 48 = 24 square cm Calculate the volume of the prism.
   Full step-by-step solution

   Step 1: Find the area of the triangular base. The base is a right triangle with legs 6 cm and 8 cm. Area of triangle = (1/2) × base × height = (1/2) × 6 × 8 = (1/2) × 48 = 24 square cm Step 2: Calculate the volume of the prism. Volume of prism = area of base × height = 24 × 15 = 360 cubic cm The answer is 360 cubic centimeters.

4. Mere is researching the distance from Earth to the star Proxima Centauri, which is approximately 4.02 × 10^13 kilometers away. She learns that a space probe can travel at a speed of 5.0 × 10^4 kilometers per hour. If the probe maintains this constant speed, how many hours would it take to reach Proxima Centauri? Express your answer in proper scientific notation. Answer: 8.04 × 10^8 Solution: Write the distance and speed. Distance = 4.02 × 10^13 km Speed = 5.0 × 10^4 km/h Use the formula time = distance ÷ speed.
   Full step-by-step solution

   Step 1: Write the distance and speed. Distance = 4.02 × 10^13 km Speed = 5.0 × 10^4 km/h Step 2: Use the formula time = distance ÷ speed. Time = (4.02 × 10^13) ÷ (5.0 × 10^4) Step 3: Divide the coefficients: 4.02 ÷ 5.0 = 0.804 Step 4: Divide the powers of 10: 10^13 ÷ 10^4 = 10^(13-4) = 10^9 Step 5: Combine: 0.804 × 10^9 Step 6: Convert to proper scientific notation (coefficient between 1 and 10): 0.804 × 10^9 = 8.04 × 10^8 The answer is 8.04 × 10^8 hours.

5. The distance from Earth to the Moon is approximately 384,400,000 meters. Express this distance in scientific notation. Answer: 3.844e8 Solution: Identify the original number: 384,400,000 Place the decimal point after the first non-zero digit: 3.844 Count how many places the decimal moved from the original position: from 384,400,000.
   Full step-by-step solution

   Step 1: Identify the original number: 384,400,000 Step 2: Place the decimal point after the first non-zero digit: 3.844 Step 3: Count how many places the decimal moved from the original position: from 384,400,000. to 3.844, the decimal moved 8 places to the left Step 4: Write as 3.844 × 10^8 Step 5: In the required format, this is written as 3.844e8 Step 6: Verify: 3.844 × 100,000,000 = 384,400,000 The answer is 3.844e8.

6. Lily is studying the distances between planets in our solar system. She learns that the average distance from Earth to Mars is approximately 2.25 × 10^8 kilometers. If a spacecraft travels at a constant speed of 1.5 × 10^4 kilometers per hour, how many hours would it take to travel from Earth to Mars? Express your answer in scientific notation. Answer: 1.5 × 10^4 Solution: Distance = 2.25 × 10^8 km Speed = 1.5 × 10^4 km/h Calculate time using the formula: time = distance ÷ speed Time = (2.25 × 10^8) ÷ (1.5 × 10^4) Divide the coefficients: 2.25 ÷ 1.5 = 1.5 Divide the powers of 10: 10^8 ÷ 10^4 = 10^(8-4) = 10^4 Combine the results: 1.5 × 10^4 The answer is 1.5 ×…
   Full step-by-step solution

   Step 1: Write down the distance and speed Distance = 2.25 × 10^8 km Speed = 1.5 × 10^4 km/h Step 2: Calculate time using the formula: time = distance ÷ speed Time = (2.25 × 10^8) ÷ (1.5 × 10^4) Step 3: Divide the coefficients: 2.25 ÷ 1.5 = 1.5 Step 4: Divide the powers of 10: 10^8 ÷ 10^4 = 10^(8-4) = 10^4 Step 5: Combine the results: 1.5 × 10^4 The answer is 1.5 × 10^4 hours.

7. (3.6 × 10⁵) ÷ (4 × 10²) = ? Answer: 900 Solution: Write the division as separate fractions: (3.6 ÷ 4) × (10⁵ ÷ 10²) Divide the coefficients: 3.6 ÷ 4 = 0.9 Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Multiply the results: 0.9 × 10³ = 900 The final answer is 900.
   Full step-by-step solution

   Step 1: Write the division as separate fractions: (3.6 ÷ 4) × (10⁵ ÷ 10²) Step 2: Divide the coefficients: 3.6 ÷ 4 = 0.9 Step 3: Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³ Step 4: Multiply the results: 0.9 × 10³ = 900 Step 5: The final answer is 900.

8. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If each unit on the grid represents 3 meters in the actual garden, what is the actual area of the garden in square meters? Answer: 216 Solution: Identify the coordinates of the rectangle's vertices. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). Find the length of the rectangle.
   Full step-by-step solution

   Step 1: Identify the coordinates of the rectangle's vertices. The vertices are (2, 1), (8, 1), (8, 5), and (2, 5). Step 2: Find the length of the rectangle. Look at the bottom side from (2, 1) to (8, 1). The x-coordinates change from 2 to 8, so the length in grid units is: 8 − 2 = 6 units. Step 3: Find the width of the rectangle. Look at the left side from (2, 1) to (2, 5). The y-coordinates change from 1 to 5, so the width in grid units is: 5 − 1 = 4 units. Step 4: Calculate the area in grid units. Area = length × width = 6 × 4 = 24 square units. Step 5: Convert grid units to actual meters. Each grid unit = 3 meters in reality. So 1 square unit = (3 meters) × (3 meters) = 9 square meters. Step 6: Find the actual area in square meters. Actual area = area in grid units × conversion factor for area = 24 × 9 = 216 square meters. Final answer: 216