Math worksheet focusing on operations with scientific notation, designed for educational use.

Let's solve each section of this worksheet step by step, explaining the reasoning and showing how to work with scientific notation.

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🔷 Section A: Multiplication and Division

We use these rules:
- Multiplication: $ (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} $
- Division: $ (a \times 10^m) \div (b \times 10^n) = (a \div b) \times 10^{m-n} $

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#### 1) $ (3 \times 10^6) \times (4 \times 10^2) $

Multiply coefficients: $ 3 \times 4 = 12 $
Add exponents: $ 6 + 2 = 8 $
So: $ 12 \times 10^8 $ → Convert to scientific notation: $ 1.2 \times 10^9 $

✔ Answer: $ \boxed{1.2 \times 10^9} $

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#### 2) $ (7 \times 10^6) \times (3 \times 10^{18}) $

$ 7 \times 3 = 21 $, $ 6 + 18 = 24 $ → $ 21 \times 10^{24} $ → $ 2.1 \times 10^{25} $

✔ Answer: $ \boxed{2.1 \times 10^{25}} $

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#### 3) $ (6 \times 10^{-11}) \times (4 \times 10^{28}) $

$ 6 \times 4 = 24 $, $ -11 + 28 = 17 $ → $ 24 \times 10^{17} $ → $ 2.4 \times 10^{18} $

✔ Answer: $ \boxed{2.4 \times 10^{18}} $

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#### 4) $ (1.2 \times 10^{-3}) \times (7 \times 10^{-10}) $

$ 1.2 \times 7 = 8.4 $, $ -3 + (-10) = -13 $ → $ 8.4 \times 10^{-13} $

✔ Answer: $ \boxed{8.4 \times 10^{-13}} $

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#### 5) $ (2.5 \times 10^9) \times (8 \times 10^{20}) $

$ 2.5 \times 8 = 20 $, $ 9 + 20 = 29 $ → $ 20 \times 10^{29} $ → $ 2.0 \times 10^{30} $

✔ Answer: $ \boxed{2.0 \times 10^{30}} $

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#### 6) $ (1.1 \times 10^{-4}) \times (9 \times 10^{13}) $

$ 1.1 \times 9 = 9.9 $, $ -4 + 13 = 9 $ → $ 9.9 \times 10^9 $

✔ Answer: $ \boxed{9.9 \times 10^9} $

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#### 7) $ (2.4 \times 10^{-8}) \div (4 \times 10^{55}) $

$ 2.4 \div 4 = 0.6 $, $ -8 - 55 = -63 $ → $ 0.6 \times 10^{-63} $ → $ 6 \times 10^{-64} $

✔ Answer: $ \boxed{6 \times 10^{-64}} $

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#### 8) $ (8.6 \times 10^{39}) \div (2 \times 10^{-20}) $

$ 8.6 \div 2 = 4.3 $, $ 39 - (-20) = 39 + 20 = 59 $ → $ 4.3 \times 10^{59} $

✔ Answer: $ \boxed{4.3 \times 10^{59}} $

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🔷 Section B: Powers, Addition, Subtraction

Use exponent rules:
- $ (a \times 10^n)^m = a^m \times 10^{n \times m} $
- For addition/subtraction, convert to same power of 10 first.

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#### 1) $ (5 \times 10^3)^2 $

$ 5^2 = 25 $, $ (10^3)^2 = 10^{6} $ → $ 25 \times 10^6 $ → $ 2.5 \times 10^7 $

✔ Answer: $ \boxed{2.5 \times 10^7} $

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#### 2) $ (7 \times 10^4)^2 $

$ 7^2 = 49 $, $ (10^4)^2 = 10^8 $ → $ 49 \times 10^8 $ → $ 4.9 \times 10^9 $

✔ Answer: $ \boxed{4.9 \times 10^9} $

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#### 3) $ (3 \times 10^{-3})^3 $

$ 3^3 = 27 $, $ (10^{-3})^3 = 10^{-9} $ → $ 27 \times 10^{-9} $ → $ 2.7 \times 10^{-8} $

✔ Answer: $ \boxed{2.7 \times 10^{-8}} $

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

First, apply negative exponent: $ \frac{1}{(2 \times 10^{-2})^2} $

Now compute $ (2 \times 10^{-2})^2 = 4 \times 10^{-4} $

So: $ \frac{1}{4 \times 10^{-4}} = \frac{1}{4} \times 10^4 = 0.25 \times 10^4 = 2.5 \times 10^3 $

✔ Answer: $ \boxed{2.5 \times 10^3} $

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#### 5) $ (9 \times 10^3) + (3 \times 10^5) $

Convert both to same exponent. Use $ 10^5 $:

$ 9 \times 10^3 = 0.09 \times 10^5 $

So: $ 0.09 \times 10^5 + 3 \times 10^5 = 3.09 \times 10^5 $

✔ Answer: $ \boxed{3.09 \times 10^5} $

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#### 6) $ (2.1 \times 10^{-4}) + (3.04 \times 10^{-5}) $

Convert to same exponent. Use $ 10^{-4} $:

$ 3.04 \times 10^{-5} = 0.304 \times 10^{-4} $

So: $ 2.1 + 0.304 = 2.404 \times 10^{-4} $

✔ Answer: $ \boxed{2.404 \times 10^{-4}} $

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#### 7) $ (7.05 \times 10^7) - (4.807 \times 10^5) $

Convert $ 4.807 \times 10^5 $ to $ 0.04807 \times 10^7 $

So: $ 7.05 - 0.04807 = 7.00193 \times 10^7 $

Round appropriately? Not required unless specified — keep as is.

✔ Answer: $ \boxed{7.00193 \times 10^7} $

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#### 8) $ (4.06 \times 10^{-6}) - (9.89 \times 10^{-7}) $

Convert to same exponent: $ 10^{-6} $

$ 9.89 \times 10^{-7} = 0.989 \times 10^{-6} $

So: $ 4.06 - 0.989 = 3.071 \times 10^{-6} $

✔ Answer: $ \boxed{3.071 \times 10^{-6}} $

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🔷 Section C: Word Problems

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#### 1) GDP of California in 2014: $ \$2.3 \times 10^{12} $

##### a) 11% from manufacturing

Calculate:
$ 11\% = 0.11 $
$ 0.11 \times 2.3 \times 10^{12} = (0.11 \times 2.3) \times 10^{12} $

$ 0.11 \times 2.3 = 0.253 $ → $ 2.53 \times 10^{-1} \times 10^{12} = 2.53 \times 10^{11} $

✔ Answer: $ \boxed{2.53 \times 10^{11}} $ US dollars

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##### b) Population increase from 1994 to 2014

- 1994: $ 3.15 \times 10^7 $
- 2014: $ 3.88 \times 10^7 $

Change: $ 3.88 \times 10^7 - 3.15 \times 10^7 = 0.73 \times 10^7 = 7.3 \times 10^6 $

Percentage increase:
$$
\frac{\text{Change}}{\text{Original}} \times 100 = \frac{7.3 \times 10^6}{3.15 \times 10^7} \times 100
$$

Simplify:
$ \frac{7.3}{31.5} \times 100 $ (since $ 10^6 / 10^7 = 10^{-1} $)

$ \frac{7.3}{31.5} \approx 0.2317 $

Then $ 0.2317 \times 100 = 23.17\% $

✔ Answer: $ \boxed{23.2\%} $ (rounded to one decimal place)

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#### 2) Radius of Jupiter: $ 7.149 \times 10^4 $ km

##### a) Circumference = $ 2\pi r $

$ C = 2 \times \pi \times 7.149 \times 10^4 $

$ \pi \approx 3.1416 $, so:

$ 2 \times 3.1416 \times 7.149 \approx 6.2832 \times 7.149 \approx 44.92 $ (approx.)

So: $ C \approx 44.92 \times 10^4 = 4.492 \times 10^5 $ km

To 3 significant figures: $ \boxed{4.49 \times 10^5} $ km

✔ Answer: $ \boxed{4.49 \times 10^5} $ km

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##### b) How many days to travel around Jupiter at 120 mph?

We have:
- Distance: $ 4.49 \times 10^5 $ km
- Speed: 120 mph
- Convert speed to km/h: $ 120 \text{ mph} \times 1.6 = 192 \text{ km/h} $

Time in hours:
$ \frac{4.49 \times 10^5}{192} \approx ? $

Divide:
$ 4.49 \times 10^5 \div 192 = \frac{449000}{192} \approx 2338.54 $ hours

Convert to days: divide by 24
$ 2338.54 \div 24 \approx 97.44 $ days

✔ Answer: $ \boxed{97.4} $ days (approx.)

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✔ Final Answers Summary

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Section A

1. $ 1.2 \times 10^9 $
2. $ 2.1 \times 10^{25} $
3. $ 2.4 \times 10^{18} $
4. $ 8.4 \times 10^{-13} $
5. $ 2.0 \times 10^{30} $
6. $ 9.9 \times 10^9 $
7. $ 6 \times 10^{-64} $
8. $ 4.3 \times 10^{59} $

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Section B

1. $ 2.5 \times 10^7 $
2. $ 4.9 \times 10^9 $
3. $ 2.7 \times 10^{-8} $
4. $ 2.5 \times 10^3 $
5. $ 3.09 \times 10^5 $
6. $ 2.404 \times 10^{-4} $
7. $ 7.00193 \times 10^7 $
8. $ 3.071 \times 10^{-6} $

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Section C

1a. $ \boxed{2.53 \times 10^{11}} $ USD
1b. $ \boxed{23.2\%} $ increase
2a. $ \boxed{4.49 \times 10^5} $ km
2b. $ \boxed{97.4} $ days

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