Worksheet on Scientific Notation and Significant Figures with Einstein illustration.

Since the user has set all options (ExplainAnswer, TeachTopic, GeneratePractice) to false, I will proceed directly to solving the problems on the worksheet and presenting the final answers in a clear, organized manner.

I will break down each section of the worksheet:

1. Convert to Scientific Notation: Move the decimal point so there is one non-zero digit to its left. Count the moves to determine the exponent.
2. Significant Figures: Count digits based on rules (non-zeros count, leading zeros don't, trailing zeros after a decimal do, etc.).
3. Convert to Decimal Form: Move the decimal point according to the exponent.
4. Addition/Subtraction: Align exponents first, then add/subtract coefficients. Adjust significant figures based on decimal places.
5. Multiplication/Division: Multiply/divide coefficients and add/subtract exponents. Adjust significant figures to match the least precise measurement.

Let's work through them systematically.

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Section 1: Convert to Scientific Notation

* 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 x 10² -> First, $3100.0 = 3.1000 \times 10^3$. Then multiply by $10^2$: $(3.1000 \times 10^3) \times 10^2 = 3.1000 \times 10^5$
* h) 0.0114 x 10⁴ -> $0.0114 = 1.14 \times 10^{-2}$. Then $(1.14 \times 10^{-2}) \times 10^4 = 1.14 \times 10^2$
* 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 x 10⁻² -> $30.0 = 3.00 \times 10^1$. Then $(3.00 \times 10^1) \times 10^{-2} = 3.00 \times 10^{-1}$
* m) 0.982 x 10⁻³ -> $0.982 = 9.82 \times 10^{-1}$. 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 x 10⁻¹ -> Already in correct form: $3.03 \times 10^{-1}$
* q) 20.4 x 10⁵ -> $20.4 = 2.04 \times 10^1$. Then $(2.04 \times 10^1) \times 10^5 = 2.04 \times 10^6$
* r) 1.29 -> Already in correct form: $1.29 \times 10^0$
* 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 x 10³ -> $450.0 = 4.500 \times 10^2$. Then $(4.500 \times 10^2) \times 10^3 = 4.500 \times 10^5$
* v) 1000 x 10⁻³ -> $1000 = 1 \times 10^3$. Then $(1 \times 10^3) \times 10^{-3} = 1 \times 10^0$

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Section 2: Determine Significant Figures

* a) 3427: All non-zero digits count. 4 sig figs.
* b) 0.00456: Leading zeros don't count. Digits 4, 5, 6 count. 3 sig figs.
* c) 123,453: All non-zero digits count. 6 sig figs.
* d) 172: All non-zero digits count. 3 sig figs.
* e) 0.000984: Leading zeros don't count. Digits 9, 8, 4 count. 3 sig figs.
* f) 0.502: Leading zero doesn't count. Zeros between non-zeros count. Digits 5, 0, 2 count. 3 sig figs.
* g) 3100.0 x 10²: In scientific notation or with a decimal point, trailing zeros count. The number part is 3100.0. Digits 3, 1, 0, 0, 0 count. 5 sig figs.
* h) 0.0114 x 10⁴: Look at the coefficient 0.0114. Leading zeros don't count. Digits 1, 1, 4 count. 3 sig figs.
* i) 107.2: Zero between non-zeros counts. Digits 1, 0, 7, 2 count. 4 sig figs.
* j) 0.0000455: Leading zeros don't count. Digits 4, 5, 5 count. 3 sig figs.
* k) 2205.2: Zero between non-zeros counts. Digits 2, 2, 0, 5, 2 count. 5 sig figs.
* l) 30.0 x 10⁻²: Coefficient is 30.0. Trailing zero after decimal counts. Digits 3, 0, 0 count. 3 sig figs.
* m) 0.982 x 10⁻³: Coefficient is 0.982. Leading zero doesn't count. Digits 9, 8, 2 count. 3 sig figs.
* n) 0.0473: Leading zeros don't count. Digits 4, 7, 3 count. 3 sig figs.
* o) 650.502: Zero between non-zeros counts. Digits 6, 5, 0, 5, 0, 2 count. 6 sig figs.
* p) 3.03 x 10⁻¹: Coefficient 3.03. Zero between non-zeros counts. Digits 3, 0, 3 count. 3 sig figs.
* q) 20.4 x 10⁵: Coefficient 20.4. Zero between non-zeros counts. Digits 2, 0, 4 count. 3 sig figs.
* r) 1.29: All non-zero digits count. 3 sig figs.
* s) 0.00565: Leading zeros don't count. Digits 5, 6, 5 count. 3 sig figs.
* t) 1362205.2: All digits count including the zero between non-zeros. 7 sig figs.
* u) 450.0 x 10³: Coefficient 450.0. Trailing zero after decimal counts. Digits 4, 5, 0, 0 count. 4 sig figs.
* v) 1000 x 10⁻³: This is ambiguous without a decimal point. Usually, trailing zeros in an integer are not significant unless specified. However, given the context of other problems like '3100.0', it's possible this implies exactness or specific precision. But strictly by standard rules for "1000", it's often considered 1 sig fig. If written as $1.000 \times 10^3$, it would be 4. Let's assume standard rules where trailing zeros in a whole number without a decimal are not significant. So, 1 sig fig. *Self-correction*: Looking at problem g ($3100.0$), they explicitly added the decimal and zero to show significance. Here, "1000" likely means just 1 sig fig.
* w) 546,000 ± 10: The uncertainty is in the tens place (the second zero). Therefore, the digits up to that place are significant. 5, 4, 6, and the first 0 are certain. The second 0 is the uncertain digit. The last 0 is not significant because the uncertainty stops at the tens place. Wait, if it's $\pm 10$, the value is known to the nearest ten. So 546,000 implies the '0' in the tens place is the estimated digit. Thus, 5, 4, 6, 0, 0 are significant. 5 sig figs.
* x) 546,000 ± 1000: Uncertainty is in the thousands place (the first zero). So 5, 4, 6 are significant. The zeros are placeholders. 3 sig figs.

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Section 3: Convert to Decimal Form

* 1.56 x 10⁴: Move decimal 4 right -> 15,600
* 0.56 x 10⁻²: Move decimal 2 left -> 0.0056
* 3.69 x 10⁻²: Move decimal 2 left -> 0.0369
* 736.9 x 10⁵: Move decimal 5 right -> 73,690,000
* 0.00259 x 10⁵: Move decimal 5 right -> 259
* 0.000459 x 10⁻¹: Move decimal 1 left -> 0.0000459
* 13.69 x 10⁻²: Move decimal 2 left -> 0.1369
* 6.9 x 10⁴: Move decimal 4 right -> 69,000
* 0.00259 x 10³: Move decimal 3 right -> 2.59
* 0.0209 x 10⁻³: Move decimal 3 left -> 0.0000209

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Section 4: Calculate (Addition/Subtraction)

Rule: Convert to same exponent, calculate, then round to the least number of decimal places in the original numbers (based on their actual values, not just the written ones).

* a) 4.53 x 10⁵ + 2.2 x 10⁶
* Convert: $4.53 \times 10^5 = 0.453 \times 10^6$
* Add: $0.453 + 2.2 = 2.653$
* Result: $2.653 \times 10^6$
* Sig Figs/Rounding: $2.2$ has 1 decimal place (tenths). $0.453$ has 3. Limit to 1 decimal place.
* Round $2.653$ to tenths -> $2.7$
* Answer: $2.7 \times 10^6$

* b) 1913.0 - 4.6 x 10³
* Convert: $4.6 \times 10^3 = 4600$
* Subtract: $1913.0 - 4600 = -2687$
* Sig Figs/Rounding: $1913.0$ is precise to tenths. $4600$ (from $4.6 \times 10^3$) is precise to hundreds place (since the 6 is in the hundreds place).
* The result must be rounded to the hundreds place.
* $-2687$ rounded to hundreds -> $-2700$
* Scientific Notation: $-2.7 \times 10^3$

* c) 2.34 x 10²⁴ + 1.92 x 10²³
* Convert: $1.92 \times 10^{23} = 0.192 \times 10^{24}$
* Add: $2.34 + 0.192 = 2.532$
* Result: $2.532 \times 10^{24}$
* Sig Figs/Rounding: $2.34$ has 2 decimal places. $0.192$ has 3. Limit to 2 decimal places.
* Round $2.532$ to hundredths -> $2.53$
* Answer: $2.53 \times 10^{24}$

* d) 2.130 x 10³ - 6.6 x 10²
* Convert: $6.6 \times 10^2 = 0.66 \times 10^3$
* Subtract: $2.130 - 0.66 = 1.470$
* Result: $1.470 \times 10^3$
* Sig Figs/Rounding: $2.130$ has 3 decimal places. $0.66$ has 2. Limit to 2 decimal places.
* Round $1.470$ to hundredths -> $1.47$
* Answer: $1.47 \times 10^3$

* e) 9.10 x 10³ + 2.2 x 10⁶
* Convert: $9.10 \times 10^3 = 0.00910 \times 10^6$
* Add: $0.00910 + 2.2 = 2.20910$
* Result: $2.20910 \times 10^6$
* Sig Figs/Rounding: $2.2$ has 1 decimal place. $0.00910$ has 5. Limit to 1 decimal place.
* Round $2.20910$ to tenths -> $2.2$
* Answer: $2.2 \times 10^6$

* f) 1113.0 - 14.6 x 10²
* Convert: $14.6 \times 10^2 = 1460$
* Subtract: $1113.0 - 1460 = -347$
* Sig Figs/Rounding: $1113.0$ is precise to tenths. $1460$ (from $14.6 \times 10^2$) is precise to tens place (the 6 is in the tens place).
* Result must be rounded to tens place.
* $-347$ rounded to tens -> $-350$
* Scientific Notation: $-3.5 \times 10^2$

* g) 6.18 x 10⁻⁴ + 4.72 x 10⁻⁴
* Convert: $6.18 \times 10^{-45} = 0.618 \times 10^{-44}$
* Add: $0.618 + 4.72 = 5.338$
* Result: $5.338 \times 10^{-44}$
* Sig Figs/Rounding: $0.618$ has 3 decimal places. $4.72$ has 2. Limit to 2 decimal places.
* Round $5.338$ to hundredths -> $5.34$
* Answer: $5.34 \times 10^{-44}$

* h) 4.25 x 10⁻³ - 1.6 x 10⁻²
* Convert: $4.25 \times 10^{-3} = 0.425 \times 10^{-2}$
* Subtract: $0.425 - 1.6 = -1.175$
* Result: $-1.175 \times 10^{-2}$
* Sig Figs/Rounding: $0.425$ has 3 decimal places. $1.6$ has 1. Limit to 1 decimal place.
* Round $-1.175$ to tenths -> $-1.2$
* Answer: $-1.2 \times 10^{-2}$

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Section 5: Calculate (Multiplication/Division)

Rule: Perform operation. Round result to the least number of significant figures in any of the original numbers.

* a) 3.95 x 10² / 1.5 x 10⁶
* Divide coefficients: $3.95 / 1.5 = 2.6333...$
* Subtract exponents: $2 - 6 = -4$
* Raw: $2.6333... \times 10^{-4}$
* Sig Figs: $3.95$ (3 sig figs), $1.5$ (2 sig figs). Limit to 2.
* Round $2.6333$ to 2 sig figs -> $2.6$
* Answer: $2.6 \times 10^{-4}$

* b) (3.5 x 10²)(6.45 x 10¹⁰)
* Multiply coefficients: $3.5 \times 6.45 = 22.575$
* Add exponents: $2 + 10 = 12$
* Raw: $22.575 \times 10^{12}$
* Adjust to sci notation: $2.2575 \times 10^{13}$
* Sig Figs: $3.5$ (2 sig figs), $6.45$ (3 sig figs). Limit to 2.
* Round $2.2575$ to 2 sig figs -> $2.3$
* Answer: $2.3 \times 10^{13}$

* c) 4.44 x 10⁷ / 2.25 x 10⁵
* Divide coefficients: $4.44 / 2.25 = 1.97333...$
* Subtract exponents: $7 - 5 = 2$
* Raw: $1.97333... \times 10^2$
* Sig Figs: $4.44$ (3 sig figs), $2.25$ (3 sig figs). Limit to 3.
* Round $1.97333$ to 3 sig figs -> $1.97$
* Answer: $1.97 \times 10^2$

* d) (4.50 x 10⁻¹²)(3.67 x 10⁻¹²)
* Multiply coefficients: $4.50 \times 3.67 = 16.515$
* Add exponents: $-12 + (-12) = -24$
* Raw: $16.515 \times 10^{-24}$
* Adjust: $1.6515 \times 10^{-23}$
* Sig Figs: $4.50$ (3 sig figs), $3.67$ (3 sig figs). Limit to 3.
* Round $1.6515$ to 3 sig figs -> $1.65$
* Answer: $1.65 \times 10^{-23}$

* e) 1.05 x 10⁻²⁶ / 4.2 x 10⁵⁶
* Divide coefficients: $1.05 / 4.2 = 0.25$
* Subtract exponents: $-26 - 56 = -82$
* Raw: $0.25 \times 10^{-82}$
* Adjust: $2.5 \times 10^{-83}$
* Sig Figs: $1.05$ (3 sig figs), $4.2$ (2 sig figs). Limit to 2.
* $2.5$ already has 2 sig figs.
* Answer: $2.5 \times 10^{-83}$

* f) (2.5 x 10⁹)(6.45 x 10⁴)
* Multiply coefficients: $2.5 \times 6.45 = 16.125$
* Add exponents: $9 + 4 = 13$
* Raw: $16.125 \times 10^{13}$
* Adjust: $1.6125 \times 10^{14}$
* Sig Figs: $2.5$ (2 sig figs), $6.45$ (3 sig figs). Limit to 2.
* Round $1.6125$ to 2 sig figs -> $1.6$
* Answer: $1.6 \times 10^{14}$

* g) 6.022 x 10²³ / 3.011 x 10⁻⁶
* Divide coefficients: $6.022 / 3.011 = 2$ exactly.
* Subtract exponents: $23 - (-56) = 23 + 56 = 79$
* Raw: $2 \times 10^{79}$
* Sig Figs: $6.022$ (4 sig figs), $3.011$ (4 sig figs). Limit to 4.
* Write 2 as $2.000$ to show 4 sig figs.
* Answer: $2.000 \times 10^{79}$

* h) (6.88 x 10²)(3.45 x 10⁻¹⁰)
* Multiply coefficients: $6.88 \times 3.45 = 23.736$
* Add exponents: $2 + (-10) = -8$
* Raw: $23.736 \times 10^{-8}$
* Adjust: $2.3736 \times 10^{-7}$
* Sig Figs: $6.88$ (3 sig figs), $3.45$ (3 sig figs). Limit to 3.
* Round $2.3736$ to 3 sig figs -> $2.37$
* Answer: $2.37 \times 10^{-7}$

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Final check of calculations and formatting. Looks good. Ready to present.

Final Answer:
Here are the solutions to the worksheet problems:

1. Convert each of the following into 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) $3.00 \times 10^{-1}$
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) $1 \times 10^0$

2. Determine the number of significant figures in each of the following:
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) 7
u) 4
v) 1
w) 5
x) 3

3. Convert each into decimal form.
$1.56 \times 10^4$ = 15,600
$0.56 \times 10^{-2}$ = 0.0056
$3.69 \times 10^{-2}$ = 0.0369
$736.9 \times 10^5$ = 73,690,000
$0.00259 \times 10^5$ = 259
$0.000459 \times 10^{-1}$ = 0.0000459
$13.69 \times 10^{-2}$ = 0.1369
$6.9 \times 10^4$ = 69,000
$0.00259 \times 10^3$ = 2.59
$0.0209 \times 10^{-3}$ = 0.0000209

4. Calculate the following. Give the answer in correct scientific notation.
a) $2.7 \times 10^6$
b) $-2.7 \times 10^3$
c) $2.53 \times 10^{24}$
d) $1.47 \times 10^3$
e) $2.2 \times 10^6$
f) $-3.5 \times 10^2$
g) $5.34 \times 10^{-44}$
h) $-1.2 \times 10^{-2}$

5. Calculate the following. Give the answer in correct scientific notation.
a) $2.6 \times 10^{-4}$
b) $2.3 \times 10^{13}$
c) $1.97 \times 10^2$
d) $1.65 \times 10^{-23}$
e) $2.5 \times 10^{-83}$
f) $1.6 \times 10^{14}$
g) $2.000 \times 10^{79}$
h) $2.37 \times 10^{-7}$