Here is the complete solution to the worksheet
“Comparing Numbers in Scientific Notation”, with explanations for each problem.
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📌 Key Rules for Comparing Numbers in Scientific Notation:
1.
Compare the exponents first.
- A larger exponent means a larger number (if positive).
- A smaller (more negative) exponent means a smaller number (if negative).
2.
If exponents are equal, compare the coefficients (the numbers before ×10^).
- Larger coefficient → larger number.
3.
Trailing zeros after a decimal do not change the value.
- e.g., `3.50 × 10⁴ = 3.5 × 10⁴`
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##
✔ Solutions & Explanations:
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1. 2.5 × 10⁶ ___ 1.8 × 10⁷
- Exponents: 6 vs 7 → 7 is larger → so 1.8 × 10⁷ is bigger.
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Answer: `<`
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2. 3 × 10⁵ ___ 6 × 10⁻⁹
- Positive exponent (5) vs negative exponent (-9) → positive is always larger.
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Answer: `>`
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3. 1 × 10¹⁵ ___ 1 × 10²⁰
- Same coefficient (1), different exponents: 15 < 20 → so 10²⁰ is bigger.
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Answer: `<`
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4. 3.5 × 10⁴ ___ 3.50 × 10⁴
- Coefficients: 3.5 = 3.50 (trailing zero doesn’t change value)
- Same exponent → equal.
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Answer: `=`
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5. 2.2 × 10⁻³¹ ___ 6.63 × 10⁻³⁴
- Exponents: -31 vs -34 → -31 is *larger* than -34 (closer to zero).
- So 10⁻³¹ > 10⁻³⁴ → even though 6.63 > 2.2, the exponent dominates.
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Answer: `>`
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6. 1.0 × 10⁻⁵ ___ 1.0 × 10⁵
- Negative exponent vs positive exponent → positive is much larger.
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Answer: `<`
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7. 1.0 × 10¹⁸ ___ 1.000 × 10¹⁸
- Coefficients: 1.0 = 1.000 (trailing zeros don’t matter)
- Same exponent → equal.
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Answer: `=`
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8. 9.8 × 10⁷ ___ 4.1 × 10⁷
- Same exponent (7) → compare coefficients: 9.8 > 4.1
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Answer: `>`
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9. 5 × 10¹⁰ ___ 7 × 10⁹
- Exponents: 10 vs 9 → 10 is larger → so 5 × 10¹⁰ > 7 × 10⁹
- Even though 7 > 5, the exponent difference (×10) makes 5 × 10¹⁰ bigger.
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Answer: `>`
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10. 7.4 × 10¹⁰ ___ 6.022 × 10²³
- Exponents: 10 vs 23 → 23 is much larger → so 6.022 × 10²³ is way bigger.
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Answer: `<`
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11. 1.25 × 10⁻⁶ ___ 1.25 × 10⁻⁶⁰
- Same coefficient → compare exponents: -6 vs -60 → -6 is larger (less negative)
- So 10⁻⁶ > 10⁻⁶⁰ → first number is larger.
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Answer: `>`
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12. 3.2 × 10⁻²⁵ ___ 9.08 × 10²⁵
- Negative exponent vs positive exponent → positive is much larger.
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Answer: `<`
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13. 3 × 10¹⁰⁰ ___ 2 × 10¹⁰¹
- Exponents: 100 vs 101 → 101 is larger → so 2 × 10¹⁰¹ is bigger.
- Even though 3 > 2, 10¹⁰¹ = 10 × 10¹⁰⁰ → so 2 × 10¹⁰¹ = 20 × 10¹⁰⁰ > 3 × 10¹⁰⁰
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Answer: `<`
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14. 5.700 × 10⁸ ___ 5.7 × 10⁸
- Coefficients: 5.700 = 5.7 → same number.
- Same exponent → equal.
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Answer: `=`
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15. 4 × 10¹ ___ 4 × 10²
- Same coefficient → compare exponents: 1 vs 2 → 2 is larger → second number is bigger.
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Answer: `<`
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16. 9.9 × 10⁻⁹⁹ ___ 7.7 × 10⁻⁷⁷
- Both negative exponents → compare: -99 vs -77 → -77 is *larger* (closer to zero)
- So 10⁻⁷⁷ > 10⁻⁹⁹ → even though 9.9 > 7.7, the exponent difference dominates.
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Answer: `<`
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## ✍️ Final Answer Key:
| Problem | Symbol |
|---------|--------|
| 1 | `<` |
| 2 | `>` |
| 3 | `<` |
| 4 | `=` |
| 5 | `>` |
| 6 | `<` |
| 7 | `=` |
| 8 | `>` |
| 9 | `>` |
| 10 | `<` |
| 11 | `>` |
| 12 | `<` |
| 13 | `<` |
| 14 | `=` |
| 15 | `<` |
| 16 | `<` |
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✔ You can now confidently fill in all the circles on your worksheet! Let me know if you want a printable version or more practice problems.
Parent Tip: Review the logic above to help your child master the concept of comparing numbers in scientific notation worksheet.
