Scientific Notation

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Scientific Notation · Grade 7 · Worksheet 1

1. Mason is researching the population of a rare species of butterfly. He finds that the current population is approximately 2.7 × 10^5 butterflies. Due to conservation efforts, the population is expected to increase by a factor of 2.2 × 10^2 over the next decade. What will be the expected population of butterflies after the increase? Express your answer in proper scientific notation. Answer: 5.94 × 10^7 Solution: Write the current population and growth factor in scientific notation. Current population = 2.7 × 10^5. Growth factor = 2.2 × 10^2.
   Full step-by-step solution

   Step 1: Write the current population and growth factor in scientific notation. Current population = 2.7 × 10^5. Growth factor = 2.2 × 10^2. Step 2: Multiply the coefficients: 2.7 × 2.2 = 5.94. Step 3: Multiply the powers of 10 by adding the exponents: 10^5 × 10^2 = 10^(5+2) = 10^7. Step 4: Combine the results: 5.94 × 10^7. Step 5: Check proper scientific notation: The coefficient 5.94 is between 1 and 10, so this is correct. The expected population is 5.94 × 10^7 butterflies.

2. Matiu is helping his class with a science project about light pollution. He learns that the city of Auckland, New Zealand, has approximately 1,620,000 residents. Write this number in proper scientific notation. Answer: 1.62 × 10^6 Solution: Start with the number 1,620,000. To write in scientific notation, we need a coefficient between 1 and 10. Move the decimal point to the left until only one non-zero digit remains to the left of the decimal: 1.620000.
   Full step-by-step solution

   Step 1: Start with the number 1,620,000. Step 2: To write in scientific notation, we need a coefficient between 1 and 10. Move the decimal point to the left until only one non-zero digit remains to the left of the decimal: 1.620000. Step 3: Count how many places we moved the decimal point. Starting from the end of 1,620,000 (which is 1,620,000.0), we move the decimal 6 places to the left to get 1.62. Step 4: The number of places moved becomes the exponent of 10. Since we moved left, the exponent is positive 6. Step 5: Write the result as 1.62 × 10^6. Step 6: Check that the coefficient is between 1 and 10 (1.62 is between 1 and 10). The answer is 1.62 × 10^6.

3. 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.

4. Aroha is looking at a diagram of a rectangular field on a coordinate grid. The field has vertices at (0, 0), (9000, 0), (9000, 4000), and (0, 4000). Each unit on the grid represents 1 meter in real life. Aroha learns that the area of the field can be written in scientific notation as a × 10ⁿ square meters, where 1 ≤ a < 10. What is the value of a in this scientific notation representation? Answer: 3.6 Solution: Find the length and width of the field in meters. The length is the horizontal distance from (0,0) to (9000,0), which is 9000 meters. The width is the vertical distance from (0,0) to (0,4000), which is 4000 meters.
   Full step-by-step solution

   Step 1: Find the length and width of the field in meters. The length is the horizontal distance from (0,0) to (9000,0), which is 9000 meters. The width is the vertical distance from (0,0) to (0,4000), which is 4000 meters. Step 2: Calculate the area of the rectangle: Area = length × width = 9000 × 4000 = 36,000,000 square meters. Step 3: Write 36,000,000 in scientific notation. Move the decimal point 7 places to the left: 36,000,000 = 3.6 × 10⁷. Step 4: In the form a × 10ⁿ, a = 3.6. The answer is 3.6.

5. A scientist is studying the growth of bacteria in a petri dish. The initial population is 2.5 × 10^4 bacteria. The population doubles every 2 hours. After 6 hours, how many bacteria are in the petri dish? Express your answer in scientific notation. Answer: 2.0 × 10^5 Solution: Identify the initial population: 2.5 × 10^4 Determine the number of doubling periods in 6 hours: 6 hours ÷ 2 hours/doubling = 3 doublings Calculate the multiplier for 3 doublings: 2 × 2 × 2 = 2^3 = 8 Multiply the initial population by the multiplier: (2.5 × 10^4) × 8 = 20.0 × 10^4 Convert to…
   Full step-by-step solution

   Step 1: Identify the initial population: 2.5 × 10^4 Step 2: Determine the number of doubling periods in 6 hours: 6 hours ÷ 2 hours/doubling = 3 doublings Step 3: Calculate the multiplier for 3 doublings: 2 × 2 × 2 = 2^3 = 8 Step 4: Multiply the initial population by the multiplier: (2.5 × 10^4) × 8 = 20.0 × 10^4 Step 5: Convert to proper scientific notation: 20.0 × 10^4 = 2.0 × 10^5 After 6 hours, there are 2.0 × 10^5 bacteria in the petri dish.

6. (4.5 × 10⁴) ÷ (9 × 10²) = ? Answer: 50 Solution: Write the division of the two numbers: (4.5 × 10⁴) ÷ (9 × 10²) Separate the coefficients and the powers of 10: (4.5 ÷ 9) × (10⁴ ÷ 10²) Divide the coefficients: 4.5 ÷ 9 = 0.5 Divide the powers of 10 by subtracting exponents: 10⁴ ÷ 10² = 10^(4-2) = 10² Multiply the results: 0.5 × 10² Convert to…
   Full step-by-step solution

   Step 1: Write the division of the two numbers: (4.5 × 10⁴) ÷ (9 × 10²) Step 2: Separate the coefficients and the powers of 10: (4.5 ÷ 9) × (10⁴ ÷ 10²) Step 3: Divide the coefficients: 4.5 ÷ 9 = 0.5 Step 4: Divide the powers of 10 by subtracting exponents: 10⁴ ÷ 10² = 10^(4-2) = 10² Step 5: Multiply the results: 0.5 × 10² Step 6: Convert to standard form: 0.5 × 100 = 50 Step 7: The final answer is 50.

7. (7.2 × 10⁶) ÷ (1.2 × 10³) = ? Answer: 6000 Solution: Separate the coefficients and powers of 10: (7.2 ÷ 1.2) × (10⁶ ÷ 10³) Divide the coefficients: 7.2 ÷ 1.2 = 6 Divide the powers of 10: 10⁶ ÷ 10³ = 10^(6-3) = 10³ Multiply the results: 6 × 10³ = 6 × 1000 = 6000 The answer is 6000.
   Full step-by-step solution

   Step 1: Separate the coefficients and powers of 10: (7.2 ÷ 1.2) × (10⁶ ÷ 10³) Step 2: Divide the coefficients: 7.2 ÷ 1.2 = 6 Step 3: Divide the powers of 10: 10⁶ ÷ 10³ = 10^(6-3) = 10³ Step 4: Multiply the results: 6 × 10³ = 6 × 1000 = 6000 The answer is 6000.

8. A right triangle is drawn on a coordinate plane with vertices at (0, 0), (12, 0), and (12, 5). If each unit on the grid represents 4 meters in real life, what is the actual length of the hypotenuse in meters? Round your answer to the nearest tenth. Answer: 52.0 Solution: Find the lengths of the legs in grid units. The horizontal leg goes from (0,0) to (12,0), so its length is 12 - 0 = 12 units. The vertical leg goes from (12,0) to (12,5), so its length is 5 - 0 = 5 units.
   Full step-by-step solution

   Step 1: Find the lengths of the legs in grid units. The horizontal leg goes from (0,0) to (12,0), so its length is 12 - 0 = 12 units. The vertical leg goes from (12,0) to (12,5), so its length is 5 - 0 = 5 units. Step 2: Use the Pythagorean theorem to find the hypotenuse in grid units. a² + b² = c² 12² + 5² = c² 144 + 25 = c² 169 = c² c = sqrt(169) = 13 units Step 3: Convert from grid units to actual meters. Each grid unit represents 4 meters, so: 13 units × 4 meters/unit = 52 meters Step 4: Round to the nearest tenth. 52.0 meters The actual length of the hypotenuse is 52.0 meters.