Scientific Operations

Grade 7 · scientific_notation · 101 practice problems · read aloud

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Scientific Notation: The Language of Science

What is Scientific Notation? 🤔

Scientific Notation is a special way to write very large or very small numbers. It makes them easier to read, compare, and calculate with! Scientists and engineers use it all the time.

A number is in scientific notation when it's written as the product of two factors: a number between 1 and 10, and a power of 10.

General Form: a × 10n (where 1 ≤ a < 10 and n is an integer)

How to Write Numbers in Scientific Notation

  1. Step 1: Move the decimal point in the original number to create a new number between 1.0 and 9.9.
  2. Step 2: Count the number of places you moved the decimal point. This becomes the exponent (n).
  3. Step 3: Write your answer as (new number) × 10n.
    • If the original number was LARGE (≥ 1), the exponent is POSITIVE.
    • If the original number was SMALL (< 1), the exponent is NEGATIVE.

Visual Examples

Example 1: Large Number - Convert 45,000,000

Step 1: Move the decimal: 4.5 (we moved it 7 places)
Step 2: Original number is large, so exponent is positive: 107
Step 3: Final answer: 4.5 × 107

Example 2: Small Number - Convert 0.0000316

Step 1: Move the decimal: 3.16 (we moved it 5 places)
Step 2: Original number is small, so exponent is negative: 10-5
Step 3: Final answer: 3.16 × 10-5

⚠️ Common Mistakes to Avoid

  • Forgetting the "× 10" part: 4.5 × 107 is correct; 4.57 is wrong!
  • Wrong exponent sign: Large numbers get positive exponents. Small numbers get negative exponents.
  • First factor not between 1 and 10: 45 × 106 is NOT in correct scientific notation.

💡 Tips & Tricks

  • L is for Large and Positive: Think "L" for Large numbers and "L" for a positive exponent (it looks like a plus sign).
  • Check your "a": Always make sure your first number is between 1 and 10.
  • Count carefully: Use your finger to count the decimal place jumps to avoid mistakes.

How to Practice

Try converting these numbers to scientific notation:

  • 6,340,000
  • 0.000802
  • The distance from Earth to the Sun is about 149,600,000 km.

Then, try converting back to standard form: 2.05 × 104 and 9.1 × 10-3.

Practice problems

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

1 (4.2 × 10⁶) ÷ (7 × 10²) = ?

Hint: Divide the coefficients first, then handle the powers of 10 separately using exponent rules.

Show the answer

Answer: 6000

  1. Write the expression: (4.2 × 10⁶) ÷ (7 × 10²)
  2. Divide the coefficients: 4.2 ÷ 7 = 0.6
  3. Divide the powers of 10: 10⁶ ÷ 10² = 10^(6-2) = 10⁴
  4. Combine the results: 0.6 × 10⁴
  5. Convert to standard form: 0.6 × 10,000 = 6,000
  6. The answer is 6000.

2 (8.5 × 10⁷) ÷ (5 × 10³) = ?

Hint: First divide the coefficients (the numbers in front), then divide the powers of ten by subtracting the exponents. Finally, adjust the result so the coefficient is between 1 and 10.

Show the answer

Answer: 17000

  1. Write the expression: (8.5 × 10⁷) ÷ (5 × 10³)
  2. Divide the coefficients: 8.5 ÷ 5 = 1.7
  3. Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴
  4. Combine: 1.7 × 10⁴
  5. Convert to standard form: 1.7 × 10000 = 17000

The answer is 17000.

3 (8.2 × 10⁷) ÷ (2 × 10³) = ?

Hint: First divide the coefficients (the numbers in front), then divide the powers of ten by subtracting the exponents. Finally, adjust the result so the coefficient is between 1 and 10.

Show the answer

Answer: 41000

  1. Write the expression: (8.2 × 10⁷) ÷ (2 × 10³)
  2. Divide the coefficients: 8.2 ÷ 2 = 4.1
  3. Divide the powers of ten: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴
  4. Combine the results: 4.1 × 10⁴
  5. Convert to standard form: 4.1 × 10000 = 41000

The answer is 41000.

4 (3.4 × 10⁵) × (2.5 × 10³) = ?

Hint: Multiply the coefficients first, then add the exponents when multiplying powers of ten.

Show the answer

Answer: 8.5 × 10⁸

  1. Separate the decimal parts and the powers of 10. We have: (3.4 × 2.5) × (10⁵ × 10³)
  2. Multiply the decimal parts. 3.4 × 2.5 = ? First, 3 × 2.5 = 7.5 Then, 0.4 × 2.5 = 1.0 Add them: 7.5 + 1.0 = 8.5 So, 3.4 × 2.5 = 8.5
  3. Multiply the powers of 10. 10⁵ × 10³ = 10^(5 + 3) = 10⁸ (When multiplying powers with the same base, add the exponents.)
  4. Combine the results. 8.5 × 10⁸
  5. Check if the decimal part is in correct scientific notation. 8.5 is between 1 and 10, so it's already in proper scientific notation. Final answer: 8.5 × 10⁸

Let's multiply (3.4 × 10⁵) × (2.5 × 10³) step by step.

5 (6.5 × 10⁷) × (4.0 × 10³) = ?

Hint: When multiplying numbers in scientific notation, multiply the coefficients together and add the exponents of the powers of ten. Then check if the coefficient is between 1 and 10.

Show the answer

Answer: 2.6 × 10¹¹

  1. Write the expression: (6.5 × 10⁷) × (4.0 × 10³)
  2. Separate the coefficients and powers of ten: (6.5 × 4.0) × (10⁷ × 10³)
  3. Multiply the coefficients: 6.5 × 4.0 = 26.0
  4. Multiply the powers of ten by adding exponents: 10⁷ × 10³ = 10^(7+3) = 10¹⁰
  5. Combine: 26.0 × 10¹⁰
  6. Adjust to proper scientific notation: 26.0 is not between 1 and 10, so rewrite 26.0 as 2.6 × 10¹, then multiply: 2.6 × 10¹ × 10¹⁰ = 2.6 × 10¹¹

The answer is 2.6 × 10¹¹.

6 (9.1 × 10⁷) ÷ (1.3 × 10³) = ?

Hint: First divide the coefficients (the numbers in front), then divide the powers of ten by subtracting the exponents. Finally, check if the result is in proper scientific notation.

Show the answer

Answer: 70000

  1. Write the expression: (9.1 × 10⁷) ÷ (1.3 × 10³)
  2. Divide the coefficients: 9.1 ÷ 1.3 = 7
  3. Divide the powers of 10: 10⁷ ÷ 10³ = 10^(7-3) = 10⁴
  4. Combine the results: 7 × 10⁴
  5. Convert to standard form: 7 × 10000 = 70000

The answer is 70000.

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