Standard Form / Scientific Notation worksheet with problems for converting numbers to standard form and expanding scientific notation.

Let's solve the problems step by step.

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1. Write the following in standard form:

We need to express each number in scientific notation, which is of the form:

> $ a \times 10^n $

where:
- $ 1 \leq a < 10 $
- $ n $ is an integer

We move the decimal point so that only one non-zero digit is to the left of the decimal, then count how many places we moved it to determine the exponent of 10.

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#### a) 374 000
Move decimal 5 places left:
$ 3.74 \times 10^5 $

✔ Answer: $ 3.74 \times 10^5 $

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#### b) 1200
Move decimal 3 places left:
$ 1.2 \times 10^3 $

✔ Answer: $ 1.2 \times 10^3 $

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#### c) 745.68
Move decimal 2 places left:
$ 7.4568 \times 10^2 $

✔ Answer: $ 7.4568 \times 10^2 $

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#### d) 29.37
Move decimal 1 place left:
$ 2.937 \times 10^1 $

✔ Answer: $ 2.937 \times 10^1 $

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#### e) 70 000
Move decimal 4 places left:
$ 7 \times 10^4 $

✔ Answer: $ 7 \times 10^4 $

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#### f) 3645.21
Move decimal 3 places left:
$ 3.64521 \times 10^3 $

✔ Answer: $ 3.64521 \times 10^3 $

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#### g) 1210.4
Move decimal 3 places left:
$ 1.2104 \times 10^3 $

✔ Answer: $ 1.2104 \times 10^3 $

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#### h) 74 006
Move decimal 4 places left:
$ 7.4006 \times 10^4 $

✔ Answer: $ 7.4006 \times 10^4 $

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#### i) 1 000
Move decimal 3 places left:
$ 1 \times 10^3 $

✔ Answer: $ 1 \times 10^3 $

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#### j) 98365
Move decimal 4 places left:
$ 9.8365 \times 10^4 $

✔ Answer: $ 9.8365 \times 10^4 $

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✔ Summary for Part 1:

| Problem | Scientific Notation |
|--------|---------------------|
| a) 374 000 | $ 3.74 \times 10^5 $ |
| b) 1200 | $ 1.2 \times 10^3 $ |
| c) 745.68 | $ 7.4568 \times 10^2 $ |
| d) 29.37 | $ 2.937 \times 10^1 $ |
| e) 70 000 | $ 7 \times 10^4 $ |
| f) 3645.21 | $ 3.64521 \times 10^3 $ |
| g) 1210.4 | $ 1.2104 \times 10^3 $ |
| h) 74 006 | $ 7.4006 \times 10^4 $ |
| i) 1 000 | $ 1 \times 10^3 $ |
| j) 98365 | $ 9.8365 \times 10^4 $ |

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2. Expand the following:

Now we convert from scientific notation back to standard form (regular number).

We multiply the coefficient by $ 10^n $, which means moving the decimal point $ n $ places to the right.

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#### a) $ 3.74 \times 10^2 $
$ 10^2 = 100 $ → $ 3.74 \times 100 = 374 $

✔ Answer: 374

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#### b) $ 8.2 \times 10^2 $
$ 8.2 \times 100 = 820 $

✔ Answer: 820

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#### c) $ 3.153 \times 10^3 $
$ 10^3 = 1000 $ → $ 3.153 \times 1000 = 3153 $

✔ Answer: 3153

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#### d) $ 4.789 \times 10^4 $
$ 10^4 = 10000 $ → $ 4.789 \times 10000 = 47890 $

✔ Answer: 47890

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#### e) $ 2.5 \times 10^2 $
$ 2.5 \times 100 = 250 $

✔ Answer: 250

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#### f) $ 1.79 \times 10^4 $
$ 1.79 \times 10000 = 17900 $

✔ Answer: 17900

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#### g) $ 9.306 \times 10^4 $
$ 9.306 \times 10000 = 93060 $

✔ Answer: 93060

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#### h) $ 5.005 \times 10^5 $
$ 10^5 = 100000 $ → $ 5.005 \times 100000 = 500500 $

✔ Answer: 500500

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#### i) $ 7 \times 10^2 $
$ 7 \times 100 = 700 $

✔ Answer: 700

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#### j) $ 6 \times 10^4 $
$ 6 \times 10000 = 60000 $

✔ Answer: 60000

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✔ Summary for Part 2:

| Problem | Expanded Form |
|--------|----------------|
| a) $ 3.74 \times 10^2 $ | 374 |
| b) $ 8.2 \times 10^2 $ | 820 |
| c) $ 3.153 \times 10^3 $ | 3153 |
| d) $ 4.789 \times 10^4 $ | 47890 |
| e) $ 2.5 \times 10^2 $ | 250 |
| f) $ 1.79 \times 10^4 $ | 17900 |
| g) $ 9.306 \times 10^4 $ | 93060 |
| h) $ 5.005 \times 10^5 $ | 500500 |
| i) $ 7 \times 10^2 $ | 700 |
| j) $ 6 \times 10^4 $ | 60000 |

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✔ Final Answers:

#### Part 1: Standard Form (Scientific Notation)

a) $ 3.74 \times 10^5 $
b) $ 1.2 \times 10^3 $
c) $ 7.4568 \times 10^2 $
d) $ 2.937 \times 10^1 $
e) $ 7 \times 10^4 $
f) $ 3.64521 \times 10^3 $
g) $ 1.2104 \times 10^3 $
h) $ 7.4006 \times 10^4 $
i) $ 1 \times 10^3 $
j) $ 9.8365 \times 10^4 $

#### Part 2: Expand (Standard Form)

a) 374
b) 820
c) 3153
d) 47890
e) 250
f) 17900
g) 93060
h) 500500
i) 700
j) 60000

Let me know if you'd like this formatted as a printable sheet or with explanations in simpler terms!