Scientific Notation

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Answer Key & Explanations

Scientific Notation · Grade 7 · Worksheet 2

1. (3.6 × 10⁸) ÷ (4.5 × 10⁴) = ? Answer: 8000 Solution: Separate the coefficients and powers of 10: (3.6 ÷ 4.5) × (10⁸ ÷ 10⁴) Divide the coefficients: 3.6 ÷ 4.5 = 0.8 Divide the powers of 10: 10⁸ ÷ 10⁴ = 10⁸⁻⁴ = 10⁴ Combine the results: 0.8 × 10⁴ Convert to standard form: 0.8 × 10,000 = 8,000 The answer is 8000.
   Full step-by-step solution

   Step 1: Separate the coefficients and powers of 10: (3.6 ÷ 4.5) × (10⁸ ÷ 10⁴) Step 2: Divide the coefficients: 3.6 ÷ 4.5 = 0.8 Step 3: Divide the powers of 10: 10⁸ ÷ 10⁴ = 10⁸⁻⁴ = 10⁴ Step 4: Combine the results: 0.8 × 10⁴ Step 5: Convert to standard form: 0.8 × 10,000 = 8,000 The answer is 8000.

2. (3.6 × 10⁸) ÷ (4.5 × 10³) = ? Answer: 80000 Solution: Separate the decimal parts and the powers of 10: (3.6 ÷ 4.5) × (10⁸ ÷ 10³) Calculate 3.6 ÷ 4.5 = 0.8 Calculate 10⁸ ÷ 10³ = 10⁵ (subtract exponents: 8 - 3 = 5) Multiply the results: 0.8 × 10⁵ = 8 × 10⁴ Convert to standard form: 8 × 10⁴ = 80,000 The answer is 80000.
   Full step-by-step solution

   Step 1: Separate the decimal parts and the powers of 10: (3.6 ÷ 4.5) × (10⁸ ÷ 10³) Step 2: Calculate 3.6 ÷ 4.5 = 0.8 Step 3: Calculate 10⁸ ÷ 10³ = 10⁵ (subtract exponents: 8 - 3 = 5) Step 4: Multiply the results: 0.8 × 10⁵ = 8 × 10⁴ Step 5: Convert to standard form: 8 × 10⁴ = 80,000 The answer is 80000.

3. Aroha is looking at a scale model of a solar system display in a museum. The model shows the distance from the Sun to Neptune as 4.5 × 10³ millimeters. The actual distance from the Sun to Neptune is 4.5 × 10¹² millimeters. On a separate diagram, the model's scale is represented by a line segment that is 9 millimeters long, labeled with the actual distance it represents. What actual distance, in millimeters, does that 9-millimeter line segment represent? Express your answer in scientific notation. Answer: 9.0 × 10⁹ Solution: The model distance from Sun to Neptune is 4.5 × 10³ mm. The actual distance is 4.5 × 10¹² mm. The scale factor from model to actual is (4.5 × 10¹²) / (4.5 × 10³) = 10⁹.
   Full step-by-step solution

   Step 1: The model distance from Sun to Neptune is 4.5 × 10³ mm. The actual distance is 4.5 × 10¹² mm. The scale factor from model to actual is (4.5 × 10¹²) / (4.5 × 10³) = 10⁹. This means 1 mm on the model represents 10⁹ mm in reality. Step 2: A 9-millimeter line segment on the model represents 9 × 10⁹ mm in actual distance. Step 3: In scientific notation, 9 × 10⁹ is already in proper form because 1 ≤ 9 < 10. The answer is 9.0 × 10⁹ millimeters.

4. A rectangular prism is drawn with dimensions: length = 12 cm, width = 8 cm, and height = 15 cm. If you were to draw a net of this prism, what would be the total surface area of all the faces combined? Answer: 792 Solution: - Two faces with dimensions 12 cm × 8 cm - Two faces with dimensions 12 cm × 15 cm - Two faces with dimensions 8 cm × 15 cm Area of 12×8 faces = 2 × (12 × 8) = 2 × 96 = 192 cm² Area of 12×15 faces = 2 × (12 × 15) = 2 × 180 = 360 cm² Area of 8×15 faces = 2 × (8 × 15) = 2 × 120 = 240 cm² Total…
   Full step-by-step solution

   Step 1: Identify the three pairs of identical faces: - Two faces with dimensions 12 cm × 8 cm - Two faces with dimensions 12 cm × 15 cm - Two faces with dimensions 8 cm × 15 cm Step 2: Calculate the area of each pair: Area of 12×8 faces = 2 × (12 × 8) = 2 × 96 = 192 cm² Area of 12×15 faces = 2 × (12 × 15) = 2 × 180 = 360 cm² Area of 8×15 faces = 2 × (8 × 15) = 2 × 120 = 240 cm² Step 3: Add all the areas together: Total surface area = 192 + 360 + 240 = 792 cm² The answer is 792.

5. Isabella is researching the population of a small town. The population was recorded as 32,700 people last year. She needs to write this number in proper scientific notation for her science report. What is 32,700 written in scientific notation? Answer: 3.27 × 10⁴ Solution: Write the number 32,700. Step 2: Move the decimal point to the left until you have a number between 1 and 10. 32,700 → 3.2700 → 3.27.
   Full step-by-step solution

   Step 1: Write the number 32,700. Step 2: Move the decimal point to the left until you have a number between 1 and 10. 32,700 → 3.2700 → 3.27. Step 3: Count how many places you moved the decimal point: 4 places to the left. Step 4: The exponent is positive 4 because you moved left. Step 5: Write the number in scientific notation: 3.27 × 10⁴. The answer is 3.27 × 10⁴.

6. Emma is looking at a diagram of a microchip. A single memory cell on the chip is a tiny square with a side length of 0.0000035 meters. What is the area of this memory cell in square meters, written in proper scientific notation? Answer: 1.225 × 10⁻¹¹ Solution: Find the area of the square. Area = side × side = 0.0000035 × 0.0000035. Write 0.0000035 in scientific notation.
   Full step-by-step solution

   Step 1: Find the area of the square. Area = side × side = 0.0000035 × 0.0000035. Step 2: Write 0.0000035 in scientific notation. The decimal point moves 6 places to the right to get 3.5, so 0.0000035 = 3.5 × 10⁻⁶. Step 3: Multiply the two numbers in scientific notation: (3.5 × 10⁻⁶) × (3.5 × 10⁻⁶). Step 4: Multiply the coefficients: 3.5 × 3.5 = 12.25. Step 5: Multiply the powers of 10: 10⁻⁶ × 10⁻⁶ = 10⁻¹². Step 6: Combine: 12.25 × 10⁻¹². This is not in proper scientific notation because 12.25 is not between 1 and 10. Step 7: Convert to proper form: 12.25 = 1.225 × 10¹, so 12.25 × 10⁻¹² = 1.225 × 10¹ × 10⁻¹² = 1.225 × 10⁻¹¹. The area of the memory cell is 1.225 × 10⁻¹¹ square meters.