Manipulating Standard Form (A) Worksheet | Fun and Engaging PDF ... - Free Printable

Let's solve each part of this worksheet step by step, explaining the reasoning and showing how to manipulate numbers in standard form (scientific notation).

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🔷 Section A: Multiplication and Division

We use the rules:
- $ (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} $
- $ (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) $
$$
= (3 \times 4) \times 10^{6+2} = 12 \times 10^8
$$
But standard form requires a number between 1 and 10:
$$
= 1.2 \times 10^9
$$

✔ Answer: $ 1.2 \times 10^9 $

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#### 2) $ (7 \times 10^6) \times (3 \times 10^{18}) $
$$
= (7 \times 3) \times 10^{6+18} = 21 \times 10^{24}
\Rightarrow 2.1 \times 10^{25}
$$

✔ Answer: $ 2.1 \times 10^{25} $

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#### 3) $ (6 \times 10^{-11}) \times (4 \times 10^{28}) $
$$
= (6 \times 4) \times 10^{-11+28} = 24 \times 10^{17} = 2.4 \times 10^{18}
$$

✔ Answer: $ 2.4 \times 10^{18} $

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#### 4) $ (1.2 \times 10^{-3}) \times (7 \times 10^{-10}) $
$$
= (1.2 \times 7) \times 10^{-3 + (-10)} = 8.4 \times 10^{-13}
$$

✔ Answer: $ 8.4 \times 10^{-13} $

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#### 5) $ (2.5 \times 10^9) \times (8 \times 10^{20}) $
$$
= (2.5 \times 8) \times 10^{9+20} = 20 \times 10^{29} = 2.0 \times 10^{30}
$$

✔ Answer: $ 2.0 \times 10^{30} $

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#### 6) $ (1.1 \times 10^{-4}) \times (9 \times 10^{13}) $
$$
= (1.1 \times 9) \times 10^{-4+13} = 9.9 \times 10^9
$$

✔ Answer: $ 9.9 \times 10^9 $

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#### 7) $ (2.4 \times 10^{-8}) \div (4 \times 10^{55}) $
$$
= (2.4 \div 4) \times 10^{-8 - 55} = 0.6 \times 10^{-63}
\Rightarrow 6 \times 10^{-64}
$$

✔ Answer: $ 6 \times 10^{-64} $

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#### 8) $ (8.6 \times 10^{39}) \div (2 \times 10^{-20}) $
$$
= (8.6 \div 2) \times 10^{39 - (-20)} = 4.3 \times 10^{59}
$$

✔ Answer: $ 4.3 \times 10^{59} $

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🔷 Section B: Powers and Addition/Subtraction

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#### 1) $ (5 \times 10^3)^2 $
$$
= 5^2 \times (10^3)^2 = 25 \times 10^6 = 2.5 \times 10^7
$$

✔ Answer: $ 2.5 \times 10^7 $

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

✔ Answer: $ 4.9 \times 10^9 $

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

✔ Answer: $ 2.7 \times 10^{-8} $

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#### 4) $ (2 \times 10^{-2})^{-2} $
$$
= 2^{-2} \times (10^{-2})^{-2} = \frac{1}{4} \times 10^4 = 0.25 \times 10^4 = 2.5 \times 10^3
$$

✔ Answer: $ 2.5 \times 10^3 $

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

Convert both to same power of 10:
- $ 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: $ 3.09 \times 10^5 $

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

Convert to same power:
- $ 2.1 \times 10^{-4} = 21 \times 10^{-5} $
- $ 21 \times 10^{-5} + 3.04 \times 10^{-5} = 24.04 \times 10^{-5} = 2.404 \times 10^{-4} $

✔ Answer: $ 2.404 \times 10^{-4} $

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

Convert:
- $ 7.05 \times 10^7 = 705 \times 10^5 $
- $ 705 \times 10^5 - 4.807 \times 10^5 = 700.193 \times 10^5 = 7.00193 \times 10^7 $

✔ Answer: $ 7.00193 \times 10^7 $

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

Convert:
- $ 4.06 \times 10^{-6} = 40.6 \times 10^{-7} $
- $ 40.6 \times 10^{-7} - 9.89 \times 10^{-7} = 30.71 \times 10^{-7} = 3.071 \times 10^{-6} $

✔ Answer: $ 3.071 \times 10^{-6} $

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🔷 Section C: Word Problems

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#### 1a) Manufacturing contribution to GDP

GDP = $ 2.3 \times 10^{12} $
11% of that = $ 0.11 \times 2.3 \times 10^{12} $

$$
= (0.11 \times 2.3) \times 10^{12} = 0.253 \times 10^{12} = 2.53 \times 10^{11}
$$

✔ Answer: $ 2.53 \times 10^{11} $ USD

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#### 1b) Percentage increase in population

Initial: $ 3.15 \times 10^7 $
Final: $ 3.88 \times 10^7 $

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

Percentage increase:
$$
\frac{7.3 \times 10^6}{3.15 \times 10^7} \times 100 = \frac{7.3}{31.5} \times 100 \approx 23.17\%
$$

✔ Answer: Approximately 23.2%

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#### 2a) Circumference of Jupiter

Radius $ r = 7.149 \times 10^4 $ km
Circumference $ C = 2\pi r $

$$
C = 2 \times \pi \times 7.149 \times 10^4 \approx 2 \times 3.1416 \times 7.149 \times 10^4
$$

First compute:
- $ 2 \times 3.1416 \approx 6.2832 $
- $ 6.2832 \times 7.149 \approx 44.96 $ (approx)

So $ C \approx 44.96 \times 10^4 = 4.496 \times 10^5 $ km

To 3 significant figures: $ 4.50 \times 10^5 $ km

✔ Answer: $ 4.50 \times 10^5 $ km

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#### 2b) Days to travel around Jupiter at 120 mph

We have:
- Distance = $ 4.50 \times 10^5 $ km
- Speed = 120 mph
- 1 mile = 1.6 km → so convert speed to km/h:

$$
120 \text{ mph} = 120 \times 1.6 = 192 \text{ km/h}
$$

Now time in hours:
$$
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{4.50 \times 10^5}{192} \approx 2343.75 \text{ hours}
$$

Convert to days:
$$
\frac{2343.75}{24} \approx 97.656 \text{ days}
$$

To nearest day: ≈ 98 days

✔ Answer: About 98 days

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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) $ 2.53 \times 10^{11} $ USD
1b) ≈ 23.2% increase
2a) $ 4.50 \times 10^5 $ km
2b) ≈ 98 days

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