Scientific Computations

Grade 8 · scientific_notation · 101 practice problems · read aloud

🔊 Listen to this explanation

Scientific Notation: Taming Giant and Tiny Numbers

🧠 What is it and Why Use It?

Scientific notation is a way to write very large or very small numbers using powers of 10. It makes numbers easier to read, compare, and compute with. Scientists and engineers use it all the time!

Format: A number between 1 and 10, multiplied by a power of 10. Example: 5,600,000 becomes 5.6 × 106.

📝 How to Write Numbers in Scientific Notation

  1. Find the Decimal Point: Imagine a decimal point at the end of the whole number.
  2. Move the Decimal: Move the decimal point so only one non-zero digit is to its left.
  3. Count the Moves: The number of places you moved the decimal becomes the exponent.
    • If you moved it left (for large numbers), the exponent is positive.
    • If you moved it right (for small numbers), the exponent is negative.
  4. Write it: Write as: (new number) × 10(exponent).

🔍 Worked Examples

Example 1: Large Number - Convert 304,500,000

  1. Original: 304,500,000.0
  2. Move decimal left 8 places: 3.04500000
  3. Exponent is +8 (moved left, large number)
  4. Answer: 3.045 × 108

Example 2: Small Number - Convert 0.00000782

  1. Original: 0.00000782
  2. Move decimal right 6 places: 0000007.82 → 7.82
  3. Exponent is -6 (moved right, small number)
  4. Answer: 7.82 × 10-6

⚠️ Common Mistakes to Avoid

  • Wrong "A" Value: The first number (A) must be between 1 and 10. Writing 34.2 × 105 is incorrect. It should be 3.42 × 106.
  • Exponent Sign Mix-up: Remember: Left for Large numbers = Positive exponent. Right for tiny (small) numbers = Negative exponent.
  • Counting Places Wrong: Carefully count every jump, including the one to get past the first digit.

💡 Tips & Tricks

  • Memory Aid: "If the number is LARGE, the exponent is POSITIVE; if the number is a small FRACTION, the exponent is NEGATIVE."
  • Check Your Work: Expand it back out. Does 4.5 × 103 equal 4,500? Yes!
  • Shortcut: For numbers greater than 1, the exponent is one less than the number of digits before the decimal. (e.g., 52,100 has 5 digits, exponent is 4).

🎯 How to Practice

Master scientific notation with these activities:

  • Convert in Both Directions: Practice converting standard numbers to scientific notation AND scientific notation back to standard form.
  • Real-World Numbers: Look up real data like distances between planets (large numbers) or sizes of bacteria (small numbers) and convert them.
  • Flashcards: Make flashcards with a standard number on one side and its scientific notation on the other.
  • Online Quizzes: Find interactive quizzes for instant feedback on your skills!

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 (4.8 × 10⁷) ÷ (1.2 × 10³) × (3.0 × 10²) = ?

Hint: Remember to handle the coefficients and exponents separately when working with scientific notation operations.

Show the answer

Answer: 1.2 × 10⁷

  1. First handle the division: (4.8 × 10⁷) ÷ (1.2 × 10³)
  2. Divide the coefficients: 4.8 ÷ 1.2 = 4
  3. Subtract the exponents: 10⁷ ÷ 10³ = 10⁽⁷⁻³⁾ = 10⁴
  4. Now we have 4 × 10⁴
  5. Multiply by (3.0 × 10²): (4 × 10⁴) × (3.0 × 10²)
  6. Multiply the coefficients: 4 × 3.0 = 12
  7. Add the exponents: 10⁴ × 10² = 10⁽⁴⁺²⁾ = 10⁶
  8. Now we have 12 × 10⁶
  9. Convert to proper scientific notation: 12 × 10⁶ = 1.2 × 10⁷

The answer is 1.2 × 10⁷.

2 (4.8 × 10⁹) ÷ (1.2 × 10⁴) × (3.0 × 10²) = ?

Hint: Work with the coefficients and exponents separately, then combine them at the end. Remember the rules for dividing and multiplying numbers in scientific notation.

Show the answer

Answer: 1.2 × 10⁸

  1. First, divide the coefficients: 4.8 ÷ 1.2 = 4
  2. Divide the powers of 10: 10⁹ ÷ 10⁴ = 10⁹⁻⁴ = 10⁵
  3. Now we have 4 × 10⁵
  4. Multiply by the next term: (4 × 10⁵) × (3.0 × 10²)
  5. Multiply the coefficients: 4 × 3.0 = 12
  6. Multiply the powers of 10: 10⁵ × 10² = 10⁵⁺² = 10⁷
  7. Combine: 12 × 10⁷ = 1.2 × 10⁸
  8. The final answer is 1.2 × 10⁸

3 (6.4 × 10⁸) ÷ (1.6 × 10²) × (5.0 × 10⁻³) = ?

Hint: When working with scientific notation, handle the coefficients and powers of 10 separately before combining them.

Show the answer

Answer: 20000

  1. First, divide the coefficients: 6.4 ÷ 1.6 = 4
  2. Next, divide the powers of 10: 10⁸ ÷ 10² = 10^(8-2) = 10⁶
  3. Now multiply by the third term: 4 × (5.0 × 10⁻³) = 20 × 10⁻³
  4. Combine with the existing power of 10: (20 × 10⁻³) × 10⁶ = 20 × 10^(⁻³⁺⁶) = 20 × 10³
  5. Convert to standard form: 20 × 10³ = 20 × 1000 = 20000

The answer is 20000.

4 (4.8 × 10⁹) ÷ (1.2 × 10³) × (5.0 × 10⁻²) = ?

Hint: When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. When multiplying, multiply the coefficients and add the exponents.

Show the answer

Answer: 200000

  1. First, divide (4.8 × 10⁹) by (1.2 × 10³) 4.8 ÷ 1.2 = 4 10⁹ ÷ 10³ = 10⁹⁻³ = 10⁶ So, (4.8 × 10⁹) ÷ (1.2 × 10³) = 4 × 10⁶
  2. Now multiply this result by (5.0 × 10⁻²) 4 × 5.0 = 20 10⁶ × 10⁻² = 10⁶⁺⁽⁻²⁾ = 10⁴ So, 4 × 10⁶ × 5.0 × 10⁻² = 20 × 10⁴
  3. Convert 20 × 10⁴ to standard form 20 × 10⁴ = 20 × 10,000 = 200,000

The answer is 200000.

5 (4.8 × 10¹²) ÷ (1.2 × 10⁴) × (3.0 × 10⁻³) = ?

Hint: When working with scientific notation, handle the coefficients and powers of 10 separately before combining them.

Show the answer

Answer: 1.2 × 10⁶

  1. Divide the coefficients: 4.8 ÷ 1.2 = 4
  2. Divide the powers of 10: 10¹² ÷ 10⁴ = 10¹²⁻⁴ = 10⁸
  3. Multiply by the third term: 4 × (3.0 × 10⁻³) = 12 × 10⁻³
  4. Combine with previous result: (4 × 10⁸) × (3.0 × 10⁻³) = 12 × 10⁸⁻³ = 12 × 10⁵
  5. Convert to proper scientific notation: 12 × 10⁵ = 1.2 × 10¹ × 10⁵ = 1.2 × 10⁶

The answer is 1.2 × 10⁶.

6 A star is 7.5 × 10⁹ km from Earth. Light travels at 3.0 × 10⁵ km/s. How many seconds does it take for light from the star to reach Earth?

Hint: Think about the relationship between distance, speed, and time. Which operation (multiplication or division) would you use to find time when you know distance and speed?

Show the answer

Answer: 25000

  1. Write the formula: time = distance ÷ speed.
  2. Substitute the values: time = (7.5 × 10⁹) ÷ (3.0 × 10⁵).
  3. Divide the coefficients: 7.5 ÷ 3.0 = 2.5.
  4. Divide the powers of 10: 10⁹ ÷ 10⁵ = 10^(9-5) = 10⁴.
  5. Combine: 2.5 × 10⁴ = 25,000 seconds.

The answer is 25000.

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