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

Let's solve each problem step-by-step, following the rules of scientific notation and ensuring all answers are in standard form (i.e., $ a \times 10^n $, where $ 1 \leq a < 10 $).

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🔷 Section A: Multiplying & Dividing with Scientific Notation

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

$$
= 8 \times 10^{2+5} = 8 \times 10^7
$$

✔ Answer: $ 8 \times 10^7 $

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

$$
= 3 \times 10^{7 + (-2)} = 3 \times 10^5
$$

✔ Answer: $ 3 \times 10^5 $

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

$$
= 2.5 \times 10^{-7 + (-11)} = 2.5 \times 10^{-18}
$$

✔ Answer: $ 2.5 \times 10^{-18} $

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

$$
= 6 \times 10^{-3 - (-12)} = 6 \times 10^{9}
$$

✔ Answer: $ 6 \times 10^9 $

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

$$
= (2 \times 3) \times 10^{4+5} = 6 \times 10^9
$$

✔ Answer: $ 6 \times 10^9 $

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

$$
= (2 \times 7) \times 10^{-2 + (-9)} = 14 \times 10^{-11}
$$

Now adjust to standard form: $ 14 = 1.4 \times 10^1 $, so:

$$
1.4 \times 10^1 \times 10^{-11} = 1.4 \times 10^{-10}
$$

✔ Answer: $ 1.4 \times 10^{-10} $

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

First, simplify: $ 56 \div 7 = 8 $, and $ 10^{-3} \div 10^0 = 10^{-3} $

So:
$$
= 8 \times 10^{-3}
$$

✔ Answer: $ 8 \times 10^{-3} $

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

Simplify:
- $ 72 \div 6 = 12 $
- $ 10^3 \div 10^{-20} = 10^{3 - (-20)} = 10^{23} $

So:
$$
= 12 \times 10^{23}
$$

Convert to standard form: $ 12 = 1.2 \times 10^1 $, so:

$$
1.2 \times 10^1 \times 10^{23} = 1.2 \times 10^{24}
$$

✔ Answer: $ 1.2 \times 10^{24} $

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✔ Section A Answers:

| Problem | Answer |
|--------|--------|
| 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} $ |

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🔷 Section B: Powers and Operations

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

Use rule: $ (a^m)^n = a^{m \times n} $

$$
= 10^{4 \times 2} = 10^8
$$

✔ Answer: $ 1 \times 10^8 $ or just $ 10^8 $

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

$$
= 10^{-3 \times 4} = 10^{-12}
$$

✔ Answer: $ 1 \times 10^{-12} $ or $ 10^{-12} $

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

$$
= 4^2 \times (10^5)^2 = 16 \times 10^{10}
$$

Now convert to standard form: $ 16 = 1.6 \times 10^1 $

$$
= 1.6 \times 10^1 \times 10^{10} = 1.6 \times 10^{11}
$$

✔ Answer: $ 1.6 \times 10^{11} $

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

$$
= 9^2 \times (10^{-3})^2 = 81 \times 10^{-6}
$$

Now: $ 81 = 8.1 \times 10^1 $, so:

$$
= 8.1 \times 10^1 \times 10^{-6} = 8.1 \times 10^{-5}
$$

✔ Answer: $ 8.1 \times 10^{-5} $

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

Need same powers of 10.

Convert $ 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: $ 2.034 \times 10^4 $

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

Convert both to same exponent.

$ 9 \times 10^{-1} = 900 \times 10^{-3} $

$ 2 \times 10^{-3} = 2 \times 10^{-3} $

Subtract:

$$
(900 - 2) \times 10^{-3} = 898 \times 10^{-3}
$$

Now convert to standard form: $ 898 = 8.98 \times 10^2 $

So:

$$
8.98 \times 10^2 \times 10^{-3} = 8.98 \times 10^{-1}
$$

✔ Answer: $ 8.98 \times 10^{-1} $

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✔ Section B Answers:

| Problem | Answer |
|--------|--------|
| 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} $ |

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🔷 Section C: Word Problems

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#### 1a) The diameter of Earth is approximately $ 0.8 \times 10^4 $ miles.
Find equatorial circumference using $ C = \pi d $, with $ \pi \approx 3 $

$$
C = 3 \times (0.8 \times 10^4) = 2.4 \times 10^4 \text{ miles}
$$

✔ Answer: $ 2.4 \times 10^4 $ miles

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

First, find total orbits in 6 years:

$$
1.5 \times 6 = 9 \text{ orbits}
$$

Each orbit = circumference = $ 2.4 \times 10^4 $ miles

Total distance:

$$
9 \times (2.4 \times 10^4) = (9 \times 2.4) \times 10^4 = 21.6 \times 10^4
$$

Convert to standard form: $ 21.6 = 2.16 \times 10^1 $

So:

$$
2.16 \times 10^1 \times 10^4 = 2.16 \times 10^5
$$

✔ Answer: $ 2.16 \times 10^5 $ miles

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✔ Section C Answers:

- a) $ 2.4 \times 10^4 $ miles
- b) $ 2.16 \times 10^5 $ miles

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✔ Final Summary of All Answers:

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#### 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} $

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#### 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} $

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#### Section C
a. $ 2.4 \times 10^4 $ miles
b. $ 2.16 \times 10^5 $ miles

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