Comprehensive math worksheet designed to help students practice operations with scientific notation, including multiplication, division, powers, and real-world word problems.

Problem Solving and Explanation

#### Section A: Multiplying & Dividing with Scientific Notation

Instructions: Calculate the following without a calculator. Make sure your answers are written properly in standard form.

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

- Step 1: Multiply the coefficients: \( 8 \times 1 = 8 \).
- Step 2: Add the exponents of 10: \( 2 + 5 = 7 \).
- Result: \( 8 \times 10^7 \).

Answer: \( 8 \times 10^7 \)

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

- Step 1: Multiply the coefficients: \( 3 \times 1 = 3 \).
- Step 2: Add the exponents of 10: \( 7 + (-2) = 5 \).
- Result: \( 3 \times 10^5 \).

Answer: \( 3 \times 10^5 \)

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

- Step 1: Multiply the coefficients: \( 2.5 \times 1 = 2.5 \).
- Step 2: Add the exponents of 10: \( -7 + (-11) = -18 \).
- Result: \( 2.5 \times 10^{-18} \).

Answer: \( 2.5 \times 10^{-18} \)

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

- Step 1: Divide the coefficients: \( 6 \div 1 = 6 \).
- Step 2: Subtract the exponents of 10: \( -3 - (-12) = -3 + 12 = 9 \).
- Result: \( 6 \times 10^9 \).

Answer: \( 6 \times 10^9 \)

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

- Step 1: Multiply the coefficients: \( 2 \times 3 = 6 \).
- Step 2: Add the exponents of 10: \( 4 + 5 = 9 \).
- Result: \( 6 \times 10^9 \).

Answer: \( 6 \times 10^9 \)

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

- Step 1: Multiply the coefficients: \( 2 \times 7 = 14 \).
- Step 2: Add the exponents of 10: \( -2 + (-9) = -11 \).
- Result: \( 14 \times 10^{-11} \).
- Step 3: Convert to standard form: \( 14 \times 10^{-11} = 1.4 \times 10^{-10} \).

Answer: \( 1.4 \times 10^{-10} \)

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

- Step 1: Divide the coefficients: \( 56 \div 7 = 8 \).
- Step 2: Subtract the exponents of 10: \( -3 - 0 = -3 \).
- Result: \( 8 \times 10^{-3} \).

Answer: \( 8 \times 10^{-3} \)

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

- Step 1: Divide the coefficients: \( 72 \div 6 = 12 \).
- Step 2: Subtract the exponents of 10: \( 3 - (-20) = 3 + 20 = 23 \).
- Result: \( 12 \times 10^{23} \).
- Step 3: Convert to standard form: \( 12 \times 10^{23} = 1.2 \times 10^{24} \).

Answer: \( 1.2 \times 10^{24} \)

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#### Section B: Additional Calculations

Instructions: Calculate the following and leave your answers in standard form.

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

- Step 1: Use the power rule \((a^m)^n = a^{m \cdot n}\): \( (10^4)^2 = 10^{4 \cdot 2} = 10^8 \).

Answer: \( 10^8 \)

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

- Step 1: Use the power rule \((a^m)^n = a^{m \cdot n}\): \( (10^{-3})^4 = 10^{-3 \cdot 4} = 10^{-12} \).

Answer: \( 10^{-12} \)

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

- Step 1: Square the coefficient: \( 4^2 = 16 \).
- Step 2: Square the exponent of 10: \( (10^5)^2 = 10^{5 \cdot 2} = 10^{10} \).
- Step 3: Combine: \( 16 \times 10^{10} \).
- Step 4: Convert to standard form: \( 16 \times 10^{10} = 1.6 \times 10^{11} \).

Answer: \( 1.6 \times 10^{11} \)

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

- Step 1: Square the coefficient: \( 9^2 = 81 \).
- Step 2: Square the exponent of 10: \( (10^{-3})^2 = 10^{-3 \cdot 2} = 10^{-6} \).
- Step 3: Combine: \( 81 \times 10^{-6} \).
- Step 4: Convert to standard form: \( 81 \times 10^{-6} = 8.1 \times 10^{-5} \).

Answer: \( 8.1 \times 10^{-5} \)

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

- Step 1: Align the powers of 10 by converting \( 3.4 \times 10^2 \) to match \( 10^4 \):
\[
3.4 \times 10^2 = 0.034 \times 10^4
\]
- Step 2: Add the coefficients:
\[
2 \times 10^4 + 0.034 \times 10^4 = (2 + 0.034) \times 10^4 = 2.034 \times 10^4
\]

Answer: \( 2.034 \times 10^4 \)

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

- Step 1: Align the powers of 10 by converting \( 9 \times 10^{-1} \) to match \( 10^{-3} \):
\[
9 \times 10^{-1} = 900 \times 10^{-3}
\]
- Step 2: Subtract the coefficients:
\[
900 \times 10^{-3} - 2 \times 10^{-3} = (900 - 2) \times 10^{-3} = 898 \times 10^{-3}
\]
- Step 3: Convert to standard form:
\[
898 \times 10^{-3} = 8.98 \times 10^{-1}
\]

Answer: \( 8.98 \times 10^{-1} \)

---

#### Section C: Word Problems

Instructions: Solve the following problems and rewrite your answers in standard form.

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##### 1) The diameter of Earth is approximately \( 0.8 \times 10^4 \) miles.
- a. What is the equatorial circumference of Earth in standard form? Approximate pi as 3.
- b. A satellite travels around Earth \( 1 \frac{1}{2} \) times each year. How many miles has the Satellite travelled after 6 years?

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###### a. Equatorial Circumference

- Formula for circumference: \( C = \pi \times d \), where \( d \) is the diameter.
- Given: \( d = 0.8 \times 10^4 \) miles, \( \pi \approx 3 \).
- Step 1: Substitute the values:
\[
C = 3 \times (0.8 \times 10^4)
\]
- Step 2: Multiply the coefficients:
\[
3 \times 0.8 = 2.4
\]
- Step 3: Combine with the power of 10:
\[
C = 2.4 \times 10^4 \text{ miles}
\]

Answer: \( 2.4 \times 10^4 \) miles

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###### b. Distance Travelled by the Satellite

- Given: The satellite travels \( 1 \frac{1}{2} = 1.5 \) times around Earth each year.
- Step 1: Calculate the distance travelled in one year:
\[
\text{Distance per year} = 1.5 \times \text{Circumference}
\]
\[
\text{Distance per year} = 1.5 \times (2.4 \times 10^4) = 3.6 \times 10^4 \text{ miles}
\]
- Step 2: Calculate the distance travelled in 6 years:
\[
\text{Total distance} = 6 \times \text{Distance per year}
\]
\[
\text{Total distance} = 6 \times (3.6 \times 10^4) = 21.6 \times 10^4 \text{ miles}
\]
- Step 3: Convert to standard form:
\[
21.6 \times 10^4 = 2.16 \times 10^5 \text{ miles}
\]

Answer: \( 2.16 \times 10^5 \) miles

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Final Answers:

- 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. \( 10^8 \)
2. \( 10^{-12} \)
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:
1. a. \( 2.4 \times 10^4 \) miles
b. \( 2.16 \times 10^5 \) miles

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
&\text{Section A: } 8 \times 10^7, 3 \times 10^5, 2.5 \times 10^{-18}, 6 \times 10^9, 6 \times 10^9, 1.4 \times 10^{-10}, 8 \times 10^{-3}, 1.2 \times 10^{24} \\
&\text{Section B: } 10^8, 10^{-12}, 1.6 \times 10^{11}, 8.1 \times 10^{-5}, 2.034 \times 10^4, 8.98 \times 10^{-1} \\
&\text{Section C: } 2.4 \times 10^4 \text{ miles}, 2.16 \times 10^5 \text{ miles}
\end{aligned}
}
\]