Multiplying and Dividing with Scientific Notation Worksheet | Fun ... - Free Printable

Here is the complete solution to all problems in Section A, Section B, and Section C, with clear explanations and answers written in standard form (scientific notation: \( a \times 10^n \), where \( 1 \leq |a| < 10 \)).

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## ✔ SECTION A: Multiplying & Dividing with Scientific Notation

1) \( (8 \times 10^2) \times 10^5 \)

- Multiply coefficients: \( 8 \times 1 = 8 \)
- Add exponents: \( 10^{2+5} = 10^7 \)
✔ Answer: \( \boxed{8 \times 10^7} \)

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2) \( (3 \times 10^7) \times 10^{-2} \)

- Coefficients: \( 3 \times 1 = 3 \)
- Exponents: \( 10^{7 + (-2)} = 10^5 \)
✔ Answer: \( \boxed{3 \times 10^5} \)

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3) \( (2.5 \times 10^{-7}) \times 10^{-11} \)

- Coefficients: \( 2.5 \times 1 = 2.5 \)
- Exponents: \( 10^{-7 + (-11)} = 10^{-18} \)
✔ Answer: \( \boxed{2.5 \times 10^{-18}} \)

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4) \( (6 \times 10^{-3}) \div 10^{-12} \)

- Division: \( 6 \div 1 = 6 \)
- Subtract exponents: \( 10^{-3 - (-12)} = 10^{9} \)
✔ Answer: \( \boxed{6 \times 10^9} \)

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5) \( (2 \times 10^4) \times (3 \times 10^5) \)

- Multiply coefficients: \( 2 \times 3 = 6 \)
- Add exponents: \( 10^{4+5} = 10^9 \)
✔ Answer: \( \boxed{6 \times 10^9} \)

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6) \( (2 \times 10^{-2}) \times (7 \times 10^{-9}) \)

- Coefficients: \( 2 \times 7 = 14 \) → Not standard form!
- Adjust: \( 14 = 1.4 \times 10^1 \)
- Combine: \( 1.4 \times 10^1 \times 10^{-11} = 1.4 \times 10^{-10} \)
✔ Answer: \( \boxed{1.4 \times 10^{-10}} \)

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7) \( (56 \times 10^{-3}) \div (7 \times 10^0) \)

- Divide coefficients: \( 56 \div 7 = 8 \)
- Subtract exponents: \( 10^{-3 - 0} = 10^{-3} \)
✔ Answer: \( \boxed{8 \times 10^{-3}} \)

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8) \( (72 \times 10^3) \div (6 \times 10^{-20}) \)

- Divide coefficients: \( 72 \div 6 = 12 \) → Not standard form!
- Adjust: \( 12 = 1.2 \times 10^1 \)
- Exponents: \( 10^{3 - (-20)} = 10^{23} \)
- Combine: \( 1.2 \times 10^1 \times 10^{23} = 1.2 \times 10^{24} \)
✔ Answer: \( \boxed{1.2 \times 10^{24}} \)

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## ✔ SECTION B: Powers and Mixed Operations

1) \( (10^4)^2 \)

- Power rule: \( (10^a)^b = 10^{a \cdot b} \)
- So, \( 10^{4 \cdot 2} = 10^8 \)
✔ Answer: \( \boxed{1 \times 10^8} \) (or just \( 10^8 \), but standard form requires coefficient)

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2) \( (10^{-3})^4 \)

- Power rule: \( 10^{-3 \cdot 4} = 10^{-12} \)
✔ Answer: \( \boxed{1 \times 10^{-12}} \)

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3) \( (4 \times 10^5)^2 \)

- Square coefficient: \( 4^2 = 16 \)
- Square power: \( (10^5)^2 = 10^{10} \)
- Combine: \( 16 \times 10^{10} \) → adjust to standard form
- \( 16 = 1.6 \times 10^1 \), so total: \( 1.6 \times 10^1 \times 10^{10} = 1.6 \times 10^{11} \)
✔ Answer: \( \boxed{1.6 \times 10^{11}} \)

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4) \( (9 \times 10^{-3})^2 \)

- Square coefficient: \( 9^2 = 81 \)
- Square power: \( (10^{-3})^2 = 10^{-6} \)
- Combine: \( 81 \times 10^{-6} \) → adjust
- \( 81 = 8.1 \times 10^1 \), so total: \( 8.1 \times 10^1 \times 10^{-6} = 8.1 \times 10^{-5} \)
✔ Answer: \( \boxed{8.1 \times 10^{-5}} \)

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5) \( (2 \times 10^4) + (3.4 \times 10^2) \)

- Need same exponent. Convert \( 3.4 \times 10^2 \) to \( 10^4 \):
- \( 3.4 \times 10^2 = 0.034 \times 10^4 \)
- Now add: \( 2 \times 10^4 + 0.034 \times 10^4 = (2 + 0.034) \times 10^4 = 2.034 \times 10^4 \)
✔ Answer: \( \boxed{2.034 \times 10^4} \)

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6) \( (9 \times 10^{-1}) - (2 \times 10^{-3}) \)

- Convert both to same exponent. Use \( 10^{-1} \):
- \( 2 \times 10^{-3} = 0.02 \times 10^{-1} \)
- Subtract: \( 9 \times 10^{-1} - 0.02 \times 10^{-1} = (9 - 0.02) \times 10^{-1} = 8.98 \times 10^{-1} \)
✔ Answer: \( \boxed{8.98 \times 10^{-1}} \)

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## ✔ SECTION C: Word Problems

1) Earth’s diameter ≈ \( 0.8 \times 10^4 \) miles

#### a) Equatorial circumference? (Use π ≈ 3)

Circumference formula: \( C = \pi \times d \)

- \( d = 0.8 \times 10^4 \)
- \( C = 3 \times (0.8 \times 10^4) = 2.4 \times 10^4 \) miles
✔ Answer (a): \( \boxed{2.4 \times 10^4} \) miles

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#### b) Satellite travels around Earth \( 1\frac{1}{2} = 1.5 \) times per year.
How many miles after 6 years?

- Distance per year = \( 1.5 \times \text{circumference} = 1.5 \times 2.4 \times 10^4 \)
- First, \( 1.5 \times 2.4 = 3.6 \)
- So, per year: \( 3.6 \times 10^4 \) miles
- In 6 years: \( 6 \times 3.6 \times 10^4 = 21.6 \times 10^4 \)
- Convert to standard form: \( 21.6 = 2.16 \times 10^1 \), so total: \( 2.16 \times 10^1 \times 10^4 = 2.16 \times 10^5 \)

✔ Answer (b): \( \boxed{2.16 \times 10^5} \) miles

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## 🎯 Final Answers Summary:

Section A:

1. \( 8 \times 10^7 \)
2. \( 3 \times 10^5 \)
3. \( 2.5 \times 10^{-18} \)
4. \( 6 \times 10^9 \)
5. \( 6 \times 10^9 \)
6. \( 1.4 \times 10^{-10} \)
7. \( 8 \times 10^{-3} \)
8. \( 1.2 \times 10^{24} \)

Section B:

1. \( 1 \times 10^8 \)
2. \( 1 \times 10^{-12} \)
3. \( 1.6 \times 10^{11} \)
4. \( 8.1 \times 10^{-5} \)
5. \( 2.034 \times 10^4 \)
6. \( 8.98 \times 10^{-1} \)

Section C:

a. \( 2.4 \times 10^4 \) miles
b. \( 2.16 \times 10^5 \) miles

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