Worksheet covering significant digits, scientific notation, and metric conversions with various practice problems.

Let's solve each section of the "Significant Digits, Scientific Notation and Metric Conversions Worksheet" step by step with explanations.

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1. Convert from scientific notation to regular notation

We move the decimal point based on the exponent of 10:
- If the exponent is positive, move the decimal to the right.
- If the exponent is negative, move the decimal to the left.

#### a. $ 9.9 \times 10^3 $
- Move decimal 3 places right:
$ 9.9 \rightarrow 9900 $

✔ Answer: 9900

#### b. $ 7.22 \times 10^1 $
- Move decimal 1 place right:
$ 7.22 \rightarrow 72.2 $

✔ Answer: 72.2

#### c. $ 1.14 \times 10^{-1} $
- Move decimal 1 place left:
$ 1.14 \rightarrow 0.114 $

✔ Answer: 0.114

#### d. $ 1.01 \times 10^{-2} $
- Move decimal 2 places left:
$ 1.01 \rightarrow 0.0101 $

✔ Answer: 0.0101

#### e. $ 8.78 \times 10^{-3} $
- Move decimal 3 places left:
$ 8.78 \rightarrow 0.00878 $

✔ Answer: 0.00878

#### f. $ 4.1 \times 10^6 $
- Move decimal 6 places right:
$ 4.1 \rightarrow 4,100,000 $

✔ Answer: 4,100,000

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2. Convert to scientific notation (round to 1 or 2 decimal places)

We write numbers in the form $ a \times 10^n $, where $ 1 \leq a < 10 $.

#### a. $ 1,100,000 $
- Count zeros: 6 → $ 1.1 \times 10^6 $

✔ Answer: $ 1.1 \times 10^6 $

#### b. $ 2,100 $
- Move decimal to get $ 2.1 $, then count places: 3 → $ 2.1 \times 10^3 $

✔ Answer: $ 2.1 \times 10^3 $

#### c. $ 0.021 $
- Move decimal 2 places right → $ 2.1 \times 10^{-2} $

✔ Answer: $ 2.1 \times 10^{-2} $

#### d. $ 749.3 $
- Move decimal to get $ 7.493 $ → round to 2 decimals: $ 7.5 \times 10^2 $

✔ Answer: $ 7.5 \times 10^2 $

#### e. $ 12,094.9 $
- Move decimal: $ 1.20949 \times 10^4 $ → round to 2 decimals: $ 1.21 \times 10^4 $

✔ Answer: $ 1.21 \times 10^4 $

#### f. $ 0.00018544 $
- Move decimal 4 places right → $ 1.8544 \times 10^{-4} $ → round to 2 decimals: $ 1.85 \times 10^{-4} $

✔ Answer: $ 1.85 \times 10^{-4} $

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3. Determine the number of significant digits

Rules:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros are not significant.
- Trailing zeros are significant only if there’s a decimal point.

#### 1) $ 4007 $
- Non-zero digits: 4,7 → zeros in between are significant → all 4 digits
✔ Answer: 4

#### 2) $ 0.00020 $
- Leading zeros: not significant.
- First non-zero digit is 2 → then zero after it → trailing zero after decimal → significant.
- So: 2 and the last zero → 2 sig figs
✔ Answer: 2

#### 3) $ 6.900 \times 10^{-4} $
- 6,9,0,0 → both trailing zeros after decimal are significant → 4 sig figs
✔ Answer: 4

#### 4) $ 6.7 \times 10^4 $
- 6 and 7 → 2 sig figs
✔ Answer: 2

#### 5) $ 190 $
- No decimal → trailing zero ambiguous → assume not significant → 2 sig figs
✔ Answer: 2

#### 6) $ 1.7 \times 10^2 $
- 1 and 7 → 2 sig figs
✔ Answer: 2

#### 7) $ 2100 $
- No decimal → trailing zeros not significant → 2 sig figs
✔ Answer: 2

#### 8) $ 5.960 \times 10^{-9} $
- 5,9,6,0 → trailing zero after decimal → significant → 4 sig figs
✔ Answer: 4

#### 9) $ 0.370 $
- Leading zero not significant; 3,7,0 → trailing zero after decimal → significant → 3 sig figs
✔ Answer: 3

#### 10) $ 91 $
- Both digits non-zero → 2 sig figs
✔ Answer: 2

#### 11) $ 5.70 \times 10^4 $
- 5,7,0 → trailing zero after decimal → significant → 3 sig figs
✔ Answer: 3

#### 12) $ 1 \times 10^3 $
- Only one digit: 1 → 1 sig fig
✔ Answer: 1

#### 13) $ 0.00010 $
- Leading zeros: not significant → first non-zero: 1 → then zero → significant → 2 sig figs
✔ Answer: 2

#### 14) $ 5078 $
- All digits non-zero → 4 sig figs
✔ Answer: 4

#### 15) $ 300 $
- No decimal → trailing zeros not significant → 1 sig fig
✔ Answer: 1

#### 16) $ 0.000400 $
- Leading zeros: no → first non-zero: 4 → two trailing zeros → significant → 3 sig figs
✔ Answer: 3

#### 17) $ 63 $
- Two non-zero digits → 2 sig figs
✔ Answer: 2

#### 18) $ 1209 $
- All digits non-zero → 4 sig figs
✔ Answer: 4

#### 19) $ 0.00020 $
- Same as #2 → 2 sig figs
✔ Answer: 2

#### 20) $ 0.04260 $
- Leading zeros: no → 4,2,6,0 → trailing zero after decimal → significant → 4 sig figs
✔ Answer: 4

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4. Convert units using metric prefixes

Use the chart:

| Prefix | Symbol | Value |
|--------|--------|-------|
| Tera | T | $10^{12}$ |
| Giga | G | $10^9$ |
| Mega | M | $10^6$ |
| Kilo | k | $10^3$ |
| Hecto | h | $10^2$ |
| Deca | da | $10^1$ |
| Base | — | $10^0$ |
| Deci | d | $10^{-1}$ |
| Centi | c | $10^{-2}$ |
| Milli | m | $10^{-3}$ |
| Micro | μ | $10^{-6}$ |
| Nano | n | $10^{-9}$ |

#### a. $ 2.75 \text{ km} = ? \text{ cm} $
- $ 1 \text{ km} = 10^3 \text{ m}, 1 \text{ m} = 10^2 \text{ cm} $
- So $ 1 \text{ km} = 10^3 \times 10^2 = 10^5 \text{ cm} $
- $ 2.75 \times 10^5 = 275,000 \text{ cm} $

✔ Answer: 275,000 cm

#### b. $ 455 \text{ cg} = ? \text{ g} $
- $ 1 \text{ cg} = 10^{-2} \text{ g} $
- $ 455 \times 10^{-2} = 4.55 \text{ g} $

✔ Answer: 4.55 g

#### c. $ 45 \text{ dm} = ? \text{ m} $
- $ 1 \text{ dm} = 0.1 \text{ m} = 10^{-1} \text{ m} $
- $ 45 \times 0.1 = 4.5 \text{ m} $

✔ Answer: 4.5 m

#### d. $ 3.5 \text{ hg} = ? \text{ g} $
- $ 1 \text{ hg} = 100 \text{ g} = 10^2 \text{ g} $
- $ 3.5 \times 100 = 350 \text{ g} $

✔ Answer: 350 g

#### e. $ 67 \text{ mm} = ? \text{ m} $
- $ 1 \text{ mm} = 10^{-3} \text{ m} $
- $ 67 \times 10^{-3} = 0.067 \text{ m} $

✔ Answer: 0.067 m

#### f. $ 0.005 \text{ kg} = ? \text{ cg} $
- $ 1 \text{ kg} = 10^3 \text{ g}, 1 \text{ g} = 10^2 \text{ cg} $
- So $ 1 \text{ kg} = 10^3 \times 10^2 = 10^5 \text{ cg} $
- $ 0.005 \times 10^5 = 500 \text{ cg} $

✔ Answer: 500 cg

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5. Convert to km

Use: $ 1 \text{ km} = 1000 \text{ m} $, etc.

#### a. $ 1600.0 \text{ m} $
- $ 1600.0 / 1000 = 1.6000 \text{ km} $

✔ Answer: 1.6000 km

#### b. $ 2050 \text{ cm} $
- $ 1 \text{ m} = 100 \text{ cm} $ → $ 2050 / 100 = 20.5 \text{ m} $
- $ 20.5 / 1000 = 0.0205 \text{ km} $

✔ Answer: 0.0205 km

#### c. $ 1.033 \text{ Mm} $
- $ 1 \text{ Mm} = 10^6 \text{ m} $
- $ 1.033 \times 10^6 \text{ m} $
- $ / 1000 = 1.033 \times 10^3 \text{ km} = 1033 \text{ km} $

✔ Answer: 1033 km

#### d. $ 245,565 \text{ mm} $
- $ 1 \text{ m} = 1000 \text{ mm} $ → $ 245,565 / 1000 = 245.565 \text{ m} $
- $ / 1000 = 0.245565 \text{ km} $

✔ Answer: 0.245565 km

#### e. $ 20,099 \text{ m} $
- $ / 1000 = 20.099 \text{ km} $

✔ Answer: 20.099 km

#### f. $ 499 \text{ m} $
- $ / 1000 = 0.499 \text{ km} $

✔ Answer: 0.499 km

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6. Convert the following

#### a. $ 0.0075 \text{ Gm} = ? \text{ km} $
- $ 1 \text{ Gm} = 10^9 \text{ m} $
- $ 0.0075 \times 10^9 = 7.5 \times 10^6 \text{ m} $
- $ / 1000 = 7.5 \times 10^3 \text{ km} = 7500 \text{ km} $

✔ Answer: 7500 km

#### b. $ 0.00091 \text{ TL} = ? \text{ ML} $
- $ 1 \text{ TL} = 10^{12} \text{ L} $
- $ 0.00091 \times 10^{12} = 9.1 \times 10^8 \text{ L} $
- $ 1 \text{ ML} = 10^6 \text{ L} $
- $ 9.1 \times 10^8 / 10^6 = 910 \text{ ML} $

✔ Answer: 910 ML

#### c. $ 0.00046 \text{ ks} = ? \text{ cs} $
- $ 1 \text{ ks} = 1000 \text{ s} $
- $ 0.00046 \times 1000 = 0.46 \text{ s} $
- $ 1 \text{ cs} = 0.01 \text{ s} $ → $ 1 \text{ s} = 100 \text{ cs} $
- $ 0.46 \times 100 = 46 \text{ cs} $

✔ Answer: 46 cs

#### d. $ 244475.3 \text{ } \mu\text{s} = ? \text{ hs} $
- $ 1 \mu\text{s} = 10^{-6} \text{ s} $
- $ 244475.3 \times 10^{-6} = 0.2444753 \text{ s} $
- $ 1 \text{ hs} = 0.1 \text{ s} $ → $ 1 \text{ s} = 10 \text{ hs} $
- $ 0.2444753 \times 10 = 2.444753 \text{ hs} $

✔ Answer: 2.444753 hs

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✔ Final Answers Summary:

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#### 1. Scientific Notation → Regular
a. 9900
b. 72.2
c. 0.114
d. 0.0101
e. 0.00878
f. 4,100,000

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#### 2. Regular → Scientific Notation
a. $ 1.1 \times 10^6 $
b. $ 2.1 \times 10^3 $
c. $ 2.1 \times 10^{-2} $
d. $ 7.5 \times 10^2 $
e. $ 1.21 \times 10^4 $
f. $ 1.85 \times 10^{-4} $

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#### 3. Significant Digits
1) 4
2) 2
3) 4
4) 2
5) 2
6) 2
7) 2
8) 4
9) 3
10) 2
11) 3
12) 1
13) 2
14) 4
15) 1
16) 3
17) 2
18) 4
19) 2
20) 4

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#### 4. Metric Conversions
- $ 2.75 \text{ km} = 275,000 \text{ cm} $
- $ 455 \text{ cg} = 4.55 \text{ g} $
- $ 45 \text{ dm} = 4.5 \text{ m} $
- $ 3.5 \text{ hg} = 350 \text{ g} $
- $ 67 \text{ mm} = 0.067 \text{ m} $
- $ 0.005 \text{ kg} = 500 \text{ cg} $

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#### 5. Convert to km
- 1600.0 m = 1.6000 km
- 2050 cm = 0.0205 km
- 1.033 Mm = 1033 km
- 245,565 mm = 0.245565 km
- 20,099 m = 20.099 km
- 499 m = 0.499 km

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#### 6. Unit Conversions
- $ 0.0075 \text{ Gm} = 7500 \text{ km} $
- $ 0.00091 \text{ TL} = 910 \text{ ML} $
- $ 0.00046 \text{ ks} = 46 \text{ cs} $
- $ 244475.3 \mu\text{s} = 2.444753 \text{ hs} $

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