Scientific Comparison

Grade 7 · scientific_notation · 100 practice problems · read aloud

🔊 Listen to this explanation

Scientific Comparison: Mastering Scientific Notation

🧠 What is Scientific Comparison?

Scientific comparison is using scientific notation to quickly compare the size of very large or very small numbers. It's incredibly useful in science and astronomy where we deal with numbers like the distance to stars (very large) or the size of atoms (very small).

📝 How to Compare Numbers in Scientific Notation

  1. Compare the exponents: The number with the larger exponent is the larger number.
  2. If exponents are equal: Compare the coefficients (the numbers in front). The larger coefficient means a larger number.

🔍 Visual Examples

Example 1: Compare 4.5 × 10⁸ and 3.2 × 10⁶

Step 1: Compare exponents: 8 vs. 6.

Step 2: 8 > 6, so 4.5 × 10⁸ > 3.2 × 10⁶.

✅ The first number is larger because its exponent is larger.

Example 2: Compare 7.1 × 10⁵ and 9.4 × 10⁵

Step 1: Compare exponents: Both are 5.

Step 2: Compare coefficients: 7.1 vs. 9.4.

Step 3: 7.1 < 9.4, so 7.1 × 10⁵ < 9.4 × 10⁵.

✅ The second number is larger because the coefficients are different but exponents are the same.

⚠️ Common Mistakes to Avoid

Mistake 1: Comparing the coefficients first. Always look at the exponent first! A small number with a big exponent (3 × 10⁹) is still much larger than a big number with a small exponent (9 × 10²).

Mistake 2: Forgetting that a negative exponent means a very small number. 10⁻⁸ is a lot smaller than 10⁻³.

💡 Tips & Tricks

Think of the exponent as a "power level": A higher power level always wins, regardless of the coefficient!

Memory Aid: "Exponents First, Coefficients Second" (EFCS).

Visualize: 10⁸ is 100,000,000 (8 zeros), while 10⁵ is 100,000 (5 zeros). This helps you see why the exponent matters most.

🎯 Practice Suggestions

  • Create flashcards with pairs of numbers in scientific notation and practice ordering them.
  • Find real-world examples (like planet distances or cell sizes) and compare them.
  • Practice converting numbers to scientific notation first, then comparing them.
  • Try ordering three or more numbers from least to greatest.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (3.2 × 10⁴) ÷ (8 × 10²) = ?

Hint: When dividing numbers in scientific notation, divide the coefficients and subtract the exponents of the powers of 10.

Show the answer

Answer: 40

  1. Break into parts** We can separate the decimal part and the power-of-ten part: = (3.2 ÷ 8) × (10⁴ ÷ 10²) **
  2. Divide the decimal numbers** 3.2 ÷ 8 = 0.4 So now we have: 0.4 × (10⁴ ÷ 10²) **
  3. Divide the powers of ten** 10⁴ ÷ 10² = 10^(4 - 2) = 10² = 100 So now we have: 0.4 × 100 **
  4. Multiply** 0.4 × 100 = 40 **Final answer:** 40

Let's solve step by step. We have: (3.2 × 10⁴) ÷ (8 × 10²) **

2 (3.5 × 10⁴) ÷ (7 × 10²) = ?

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

Show the answer

Answer: 50

  1. Write it as a fraction. = (3.5 × 10⁴) / (7 × 10²)
  2. Separate the numbers and the powers of 10. = (3.5 / 7) × (10⁴ / 10²)
  3. Simplify 3.5 / 7. 3.5 divided by 7 is 0.5. So we have: 0.5 × (10⁴ / 10²)
  4. Simplify 10⁴ / 10². When dividing powers with the same base, subtract the exponents: 10^(4 - 2) = 10². So we have: 0.5 × 10²
  5. Multiply 0.5 by 10². 0.5 × 100 = 50 Final answer: 50

Let's solve step-by-step. We have: (3.5 × 10⁴) ÷ (7 × 10²)

3 Compare: 7.2×10⁷ __ 8.9×10⁶

Hint: Look at the exponent of 10 first. A larger exponent means the number is much larger, even if the coefficient is smaller.

Show the answer

Answer: >

  1. Write both numbers in standard form to compare. 7.2×10⁷ = 72,000,000 8.9×10⁶ = 8,900,000
  2. Compare the two numbers: 72,000,000 > 8,900,000.
  3. Therefore, 7.2×10⁷ > 8.9×10⁶.

The answer is >.

4 Compare: 7.2×10⁷ __ 8.7×10⁶

Hint: Look at the exponent first. A larger exponent means a much larger number, even if the coefficient is smaller.

Show the answer

Answer: >

  1. Write both numbers in standard form to compare. 7.2×10⁷ = 72,000,000. 8.7×10⁶ = 8,700,000.
  2. Compare the values: 72,000,000 is greater than 8,700,000.
  3. Therefore, 7.2×10⁷ > 8.7×10⁶.

The answer is >.

5 Compare: 9.3×10⁷ __ 1.5×10⁸

Hint: When comparing numbers in scientific notation, first look at the exponent of 10. The number with the larger exponent is greater, regardless of the coefficient.

Show the answer

Answer: <

  1. Identify the exponents. 9.3×10⁷ has an exponent of 7, and 1.5×10⁸ has an exponent of 8.
  2. Since 8 > 7, the number 1.5×10⁸ is larger than 9.3×10⁷.
  3. Therefore, 9.3×10⁷ is less than 1.5×10⁸.

The answer is <.

6 Compare: 9.2×10⁷ __ 1.3×10⁸

Hint: When comparing numbers in scientific notation, first look at the exponent. A larger exponent means a much larger number, even if the coefficient is smaller.

Show the answer

Answer: <

  1. Identify the exponents. 9.2×10⁷ has an exponent of 7, and 1.3×10⁸ has an exponent of 8.
  2. Since 8 > 7, the number with exponent 8 is larger, regardless of the coefficients.
  3. Therefore, 9.2×10⁷ is less than 1.3×10⁸.
  4. The correct symbol is <.

The answer is <.

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