Worksheet with exercises on scientific notation and significant figures, including conversion, determining significant digits, and calculations.

Let's solve each part of this worksheet step by step.

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#1. Convert each of the following into scientific notation.

Scientific notation format: $ a \times 10^n $, where $ 1 \leq |a| < 10 $ and $ n $ is an integer.

#### a) 3427
Move decimal 3 places left: $ 3.427 \times 10^3 $

#### b) 0.00456
Move decimal 3 places right: $ 4.56 \times 10^{-3} $

#### c) 123,453
Move decimal 5 places left: $ 1.23453 \times 10^5 $

#### d) 172
Move decimal 2 places left: $ 1.72 \times 10^2 $

#### e) 0.000984
Move decimal 4 places right: $ 9.84 \times 10^{-4} $

#### f) 0.502
Move decimal 1 place right: $ 5.02 \times 10^{-1} $

#### g) $ 3100.0 \times 10^2 $
First write 3100.0 in scientific notation: $ 3.1000 \times 10^3 $, then multiply by $ 10^2 $:
$ 3.1000 \times 10^{3+2} = 3.1000 \times 10^5 $

#### h) $ 0.0114 \times 10^4 $
Convert 0.0114 to scientific: $ 1.14 \times 10^{-2} $, then multiply:
$ 1.14 \times 10^{-2} \times 10^4 = 1.14 \times 10^{2} $

#### i) 107.2
Move decimal 2 places left: $ 1.072 \times 10^2 $

#### j) 0.0000455
Move decimal 5 places right: $ 4.55 \times 10^{-5} $

#### k) 2205.2
Move decimal 3 places left: $ 2.2052 \times 10^3 $

#### l) $ 30.0 \times 10^2 $
30.0 → $ 3.00 \times 10^1 $, so $ 3.00 \times 10^1 \times 10^2 = 3.00 \times 10^3 $

#### m) $ 0.982 \times 10^{-3} $
Convert 0.982 to $ 9.82 \times 10^{-1} $, then $ 9.82 \times 10^{-1} \times 10^{-3} = 9.82 \times 10^{-4} $

#### n) 0.0473
Move decimal 2 places right: $ 4.73 \times 10^{-2} $

#### o) 650.502
Move decimal 2 places left: $ 6.50502 \times 10^2 $

#### p) $ 3.03 \times 10^{-1} $
Already in scientific notation.

#### q) $ 20.4 \times 10^5 $
20.4 → $ 2.04 \times 10^1 $, so $ 2.04 \times 10^1 \times 10^5 = 2.04 \times 10^6 $

#### r) 1.29
Already in scientific notation: $ 1.29 \times 10^0 $

#### s) 0.00565
Move decimal 3 places right: $ 5.65 \times 10^{-3} $

#### t) 1362205.2
Move decimal 6 places left: $ 1.3622052 \times 10^6 $

#### u) $ 450.0 \times 10^3 $
450.0 → $ 4.500 \times 10^2 $, so $ 4.500 \times 10^2 \times 10^3 = 4.500 \times 10^5 $

#### v) $ 1000 \times 10^{-3} $
1000 = $ 1 \times 10^3 $, so $ 1 \times 10^3 \times 10^{-3} = 1 \times 10^0 = 1 $

But written as scientific notation: $ 1.000 \times 10^0 $ (if preserving significant figures)

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#2. Determine the number of significant figures in each of the following:

Rules:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros (before first non-zero) are not significant.
- Trailing zeros after decimal are significant.
- Trailing zeros in whole numbers without decimal are ambiguous unless specified.
- In scientific notation, all digits in coefficient are significant.

#### a) 3427 → 4 sig figs
All non-zero → 4

#### b) 0.00456 → 3 sig figs
Leading zeros don’t count → 4,5,6 → 3

#### c) 123,453 → 6 sig figs
All digits → 6

#### d) 172 → 3 sig figs
All non-zero → 3

#### e) 0.000984 → 3 sig figs
Leading zeros ignored → 9,8,4 → 3

#### f) 0.502 → 3 sig figs
Leading zero ignored, trailing zero after decimal counts → 5,0,2 → 3

#### g) $ 3100.0 \times 10^2 $ → 5 sig figs
3100.0 has 5 sig figs (trailing zero after decimal), exponent doesn't affect sig figs.

#### h) $ 0.0114 \times 10^4 $ → 3 sig figs
0.0114 → 1,1,4 → 3

#### i) 107.2 → 4 sig figs
All digits → 1,0,7,2 → 4 (zero between non-zeros counts)

#### j) 0.0000455 → 3 sig figs
4,5,5 → 3

#### k) 2205.2 → 5 sig figs
All digits → 5

#### l) $ 30.0 \times 10^2 $ → 3 sig figs
30.0 → 3 sig figs (trailing zero after decimal counts)

#### m) $ 0.982 \times 10^{-3} $ → 3 sig figs
0.982 → 9,8,2 → 3

#### n) 0.0473 → 3 sig figs
4,7,3 → 3

#### o) 650.502 → 6 sig figs
All digits → 6

#### p) $ 3.03 \times 10^{-1} $ → 3 sig figs
3,0,3 → 3

#### q) $ 20.4 \times 10^5 $ → 3 sig figs
20.4 → 3 sig figs

#### r) 1.29 → 3 sig figs

#### s) 0.00565 → 3 sig figs
5,6,5 → 3

#### t) 1362205.2 → 8 sig figs
All digits → 8

#### u) $ 450.0 \times 10^3 $ → 4 sig figs
450.0 → 4 sig figs

#### v) $ 1000 \times 10^{-3} $ → 1 sig fig?
1000 has ambiguous sig figs — no decimal → assumed 1 sig fig (but could be 4 if specified).
But here: 1000 × 10⁻³ = 1 → $ 1 \times 10^0 $ → only one digit → 1 sig fig

#### w) $ 546,000 \pm 10 $ → 3 sig figs
Uncertainty indicates precision to tens place → 546,000 ± 10 means last two digits uncertain → 546,000 → 3 sig figs (5,4,6)

#### x) $ 546,000 \pm 1000 $ → 3 sig figs
Uncertainty ±1000 → precise to thousands → 546,000 → 3 sig figs

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#3. Convert each into decimal form.

#### a) $ 1.56 \times 10^4 $ → 15600
Move decimal 4 places right: 15600

#### b) $ 0.56 \times 10^2 $ → 56
0.56 × 100 = 56

#### c) $ 3.69 \times 10^2 $ → 369
3.69 × 100 = 369

#### d) $ 736.9 \times 10^5 $ → 73,690,000
736.9 × 100,000 = 73,690,000

#### e) $ 0.00259 \times 10^5 $ → 259
0.00259 × 100,000 = 259

#### f) $ 0.000459 \times 10^{-1} $ → 0.0000459
0.000459 ÷ 10 = 0.0000459

#### g) $ 13.69 \times 10^{-3} $ → 0.01369
13.69 ÷ 1000 = 0.01369

#### h) $ 6.9 \times 10^4 $ → 69,000
6.9 × 10,000 = 69,000

#### i) $ 0.00259 \times 10^{-3} $ → 0.00000259
0.00259 ÷ 1000 = 0.00000259

#### j) $ 0.0209 \times 10^{-5} $ → 0.0000000209
0.0209 ÷ 100,000 = 0.0000000209

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#4. Calculate the following. Give answer in correct scientific notation.

#### a) $ 4.53 \times 10^5 + 2.2 \times 10^6 $

Convert to same exponent:
$ 4.53 \times 10^5 = 0.453 \times 10^6 $
Now: $ 0.453 \times 10^6 + 2.2 \times 10^6 = 2.653 \times 10^6 $

Rounded to least precise: 2.2 has 2 sig figs, 4.53 has 3 → sum should have 2 sig figs after decimal? But addition depends on decimal places.

Actually:
2.2 × 10⁶ has uncertainty in hundred thousands (±0.1×10⁶), 4.53×10⁵ is 0.453×10⁶ → both to nearest 0.1×10⁶
So sum: 2.653 × 10⁶ → round to tenths: 2.7 × 10⁶

#### b) $ 1913.0 - 4.6 \times 10^3 $

4.6 × 10³ = 4600
1913.0 - 4600 = -2687 → $ -2.687 \times 10^3 $

Significant figures: 1913.0 has 5 sig figs, 4.6×10³ has 2 sig figs → subtraction: precision limited by least precise → 4600 is known to nearest 100 → so answer to nearest 100?

1913.0 → ±0.5, 4600 → ±50 → difference ≈ ±50 → round to nearest 100:
-2687 → -2.7 × 10³

#### c) $ 2.34 \times 10^{24} + 1.92 \times 10^{23} $

Convert: $ 1.92 \times 10^{23} = 0.192 \times 10^{24} $
Sum: $ 2.34 + 0.192 = 2.532 \times 10^{24} $

Precision: 2.34 has 2 decimal places in coefficient (implied precision), 1.92 has 2 → so sum: 2.53 × 10²⁴ (round to 3 sig figs? Wait: 2.34 has 3 sig figs, 1.92 has 3 → but exponents differ by 1 → alignment matters.

But since 2.34×10²⁴ is larger, and we’re adding 0.192×10²⁴ → total 2.532×10²⁴ → round to nearest 0.01×10²⁴ → 2.53 × 10²⁴ (3 sig figs)

#### d) $ 2.130 \times 10^5 - 6.6 \times 10^2 $

6.6 × 10² = 660
2.130 × 10⁵ = 213,000
Subtract: 213,000 - 660 = 212,340 → $ 2.1234 \times 10^5 $

Now, precision: 2.130×10⁵ is precise to 100 (last digit is hundreds), 660 is precise to 10 → difference precise to 100 → round to nearest 100:
212,340 → 212,300 → $ 2.123 \times 10^5 $

But 2.130×10⁵ implies precision to 100 → so answer should be rounded to nearest 100 → 212,300 → $ 2.123 \times 10^5 $

#### e) $ 9.10 \times 10^3 + 2.2 \times 10^9 $

9.10 × 10³ = 9100
2.2 × 10⁹ = 2,200,000,000
Add: ~2,200,009,100 → $ 2.2000091 \times 10^9 $

But 2.2×10⁹ has 2 sig figs → so result must be rounded to 2 sig figs:
→ $ 2.2 \times 10^9 $

#### f) $ 1113.0 - 14.6 \times 10^2 $

14.6 × 10² = 1460
1113.0 - 1460 = -347 → $ -3.47 \times 10^2 $

Precision: 1113.0 → ±0.5, 1460 → ±5 → difference precise to ±5 → so answer to nearest 10 → -350 → $ -3.5 \times 10^2 $

#### g) $ 6.18 \times 10^{45} + 4.72 \times 10^{44} $

Convert: $ 4.72 \times 10^{44} = 0.472 \times 10^{45} $
Sum: $ 6.18 + 0.472 = 6.652 \times 10^{45} $

Both have 3 sig figs → sum: 6.65 × 10⁴⁵ (rounded to 3 sig figs)

#### h) $ 4.25 \times 10^3 - 1.6 \times 10^2 $

4.25 × 10³ = 4250
1.6 × 10² = 160
Difference: 4250 - 160 = 4090 → $ 4.09 \times 10^3 $

Precision: 4.25×10³ → ±5, 1.6×10² → ±0.05×10² = ±5 → so difference precise to ±10 → round to nearest 10 → 4090 → $ 4.09 \times 10^3 $

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#5. Calculate the following. Give answer in correct scientific notation.

Use rules for multiplication/division: sig figs based on least precise.

#### a) $ \frac{3.95 \times 10^7}{1.5 \times 10^6} = \frac{3.95}{1.5} \times 10^{7-6} = 2.633... \times 10^1 $

3.95 has 3 sig figs, 1.5 has 2 → answer: 2.6 × 10¹

#### b) $ (3.5 \times 10^2)(6.45 \times 10^{10}) = (3.5 \times 6.45) \times 10^{12} = 22.575 \times 10^{12} $

= $ 2.2575 \times 10^{13} $

3.5 has 2 sig figs → round to 2 sig figs: 2.3 × 10¹³

#### c) $ \frac{4.44 \times 10^7}{2.25 \times 10^5} = \frac{4.44}{2.25} \times 10^{2} = 1.973... \times 10^2 $

4.44 has 3, 2.25 has 3 → 1.97 × 10²

#### d) $ (4.50 \times 10^{-12})(3.67 \times 10^{-12}) = (4.50 \times 3.67) \times 10^{-24} = 16.515 \times 10^{-24} $

= $ 1.6515 \times 10^{-23} $

4.50 has 3 sig figs, 3.67 has 3 → 1.65 × 10⁻²³

#### e) $ \frac{1.05 \times 10^{26}}{4.2 \times 10^6} = \frac{1.05}{4.2} \times 10^{20} = 0.25 \times 10^{20} = 2.5 \times 10^{19} $

1.05 has 3 sig figs, 4.2 has 2 → 2.5 × 10¹⁹ (2 sig figs)

#### f) $ (2.5 \times 10^9)(6.45 \times 10^4) = 2.5 \times 6.45 \times 10^{13} = 16.125 \times 10^{13} = 1.6125 \times 10^{14} $

2.5 has 2 sig figs → 1.6 × 10¹⁴

#### g) $ \frac{6.022 \times 10^{23}}{3.011 \times 10^{-56}} = \frac{6.022}{3.011} \times 10^{23 - (-56)} = 1.999... \times 10^{79} $

≈ $ 2.000 \times 10^{79} $

6.022 has 4, 3.011 has 4 → 2.000 × 10⁷⁹

#### h) $ (6.88 \times 10^2)(3.45 \times 10^{-10}) = 6.88 \times 3.45 \times 10^{-8} = 23.624 \times 10^{-8} = 2.3624 \times 10^{-7} $

6.88 has 3, 3.45 has 3 → 2.36 × 10⁻⁷

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✔ Final Answers Summary:

#### #1 Scientific Notation:
a) $ 3.427 \times 10^3 $
b) $ 4.56 \times 10^{-3} $
c) $ 1.23453 \times 10^5 $
d) $ 1.72 \times 10^2 $
e) $ 9.84 \times 10^{-4} $
f) $ 5.02 \times 10^{-1} $
g) $ 3.1000 \times 10^5 $
h) $ 1.14 \times 10^2 $
i) $ 1.072 \times 10^2 $
j) $ 4.55 \times 10^{-5} $
k) $ 2.2052 \times 10^3 $
l) $ 3.00 \times 10^3 $
m) $ 9.82 \times 10^{-4} $
n) $ 4.73 \times 10^{-2} $
o) $ 6.50502 \times 10^2 $
p) $ 3.03 \times 10^{-1} $
q) $ 2.04 \times 10^6 $
r) $ 1.29 \times 10^0 $
s) $ 5.65 \times 10^{-3} $
t) $ 1.3622052 \times 10^6 $
u) $ 4.500 \times 10^5 $
v) $ 1.000 \times 10^0 $

#### #2 Significant Figures:
a) 4
b) 3
c) 6
d) 3
e) 3
f) 3
g) 5
h) 3
i) 4
j) 3
k) 5
l) 3
m) 3
n) 3
o) 6
p) 3
q) 3
r) 3
s) 3
t) 8
u) 4
v) 1
w) 3
x) 3

#### #3 Decimal Form:
a) 15600
b) 56
c) 369
d) 73,690,000
e) 259
f) 0.0000459
g) 0.01369
h) 69,000
i) 0.00000259
j) 0.0000000209

#### #4 Calculations:
a) $ 2.7 \times 10^6 $
b) $ -2.7 \times 10^3 $
c) $ 2.53 \times 10^{24} $
d) $ 2.123 \times 10^5 $
e) $ 2.2 \times 10^9 $
f) $ -3.5 \times 10^2 $
g) $ 6.65 \times 10^{45} $
h) $ 4.09 \times 10^3 $

#### #5 Calculations:
a) $ 2.6 \times 10^1 $
b) $ 2.3 \times 10^{13} $
c) $ 1.97 \times 10^2 $
d) $ 1.65 \times 10^{-23} $
e) $ 2.5 \times 10^{19} $
f) $ 1.6 \times 10^{14} $
g) $ 2.000 \times 10^{79} $
h) $ 2.36 \times 10^{-7} $

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