Sig.-Figs.-Sci.-Notation-Worksheet-Answer-Key - Flipbook by Foton ... - Free Printable

Let's solve each part of this Scientific Notation / Significant Figures worksheet step by step. I'll go through all problems, explain the concepts, and provide correct answers.

---

#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 to get one digit before it:
3.427 × 10³

✔ Answer: $3.427 \times 10^3$

---

#### b) 0.00456
Move decimal right:
4.56 × 10⁻³

✔ Answer: $4.56 \times 10^{-3}$

---

#### c) 123,453
→ 1.23453 × 10⁵

✔ Answer: $1.23453 \times 10^5$

---

#### d) 172
→ 1.72 × 10²

✔ Answer: $1.72 \times 10^2$

---

#### e) 0.000984
→ 9.84 × 10⁻⁴

✔ Answer: $9.84 \times 10^{-4}$

---

#### f) 0.502
→ 5.02 × 10⁻¹

✔ Answer: $5.02 \times 10^{-1}$

---

#### g) 3100.0 × 10²
First, convert 3100.0 → 3.1000 × 10³
Then multiply by 10² → 3.1000 × 10⁵

✔ Answer: $3.1000 \times 10^5$

---

#### h) 0.0114 × 10⁴
Convert 0.0114 → 1.14 × 10⁻²
Then × 10⁴ → 1.14 × 10²

✔ Answer: $1.14 \times 10^2$

---

#### i) 107.2
→ 1.072 × 10²

✔ Answer: $1.072 \times 10^2$

---

#### j) 0.0000455
→ 4.55 × 10⁻⁵

✔ Answer: $4.55 \times 10^{-5}$

---

#### k) 2205.2
→ 2.2052 × 10³

✔ Answer: $2.2052 \times 10^3$

---

#### l) 30.0 × 10⁻²
30.0 = 3.00 × 10¹ → × 10⁻² → 3.00 × 10⁻¹

✔ Answer: $3.00 \times 10^{-1}$

---

#### m) 0.982 × 10⁻³
= 9.82 × 10⁻⁴

✔ Answer: $9.82 \times 10^{-4}$

---

#### n) 0.0473
→ 4.73 × 10⁻²

✔ Answer: $4.73 \times 10^{-2}$

---

#### o) 650.502
→ 6.50502 × 10²

✔ Answer: $6.50502 \times 10^2$

---

#### p) 3.03 × 10⁻¹
Already in scientific notation.

✔ Answer: $3.03 \times 10^{-1}$

---

#### q) 20.4 × 10⁵
= 2.04 × 10⁶

✔ Answer: $2.04 \times 10^6$

---

#### r) 1.29
Already in scientific notation.

✔ Answer: $1.29 \times 10^0$ (or just 1.29)

---

#### s) 0.00565
→ 5.65 × 10⁻³

✔ Answer: $5.65 \times 10^{-3}$

---

#### t) 1362205.2
→ 1.3622052 × 10⁶

✔ Answer: $1.3622052 \times 10^6$

---

#### u) 450.0 × 10³
= 4.500 × 10⁵

✔ Answer: $4.500 \times 10^5$

---

#### v) 1000 × 10⁻³
= 1.000 × 10⁰ = 1.000

✔ Answer: $1.000 \times 10^0$

---

✔ Summary for #1:

| Letter | 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) | $1.000 \times 10^0$ |

---

#2. Determine the number of significant figures

Rules:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros are not significant.
- Trailing zeros are significant if there’s a decimal point.
- In scientific notation, only the coefficient matters.

---

#### a) 3427 → 4 sig figs (all non-zero)

✔ 4

---

#### b) 0.00456 → leading zeros not significant → 456 → 3 sig figs

✔ 3

---

#### c) 123,453 → all digits non-zero → 6 sig figs

✔ 6

---

#### d) 172 → 3 sig figs

✔ 3

---

#### e) 0.000984 → 984 → 3 sig figs

✔ 3

---

#### f) 0.502 → 5,0,2 → zero is between non-zeros → 3 sig figs

✔ 3

---

#### g) 3100.0 × 10² → 3100.0 has 5 sig figs (trailing zero after decimal counts), ×10² doesn’t change that

✔ 5

---

#### h) 0.0114 × 10⁴ → coefficient is 0.0114 → 114 → 3 sig figs

✔ 3

---

#### i) 107.2 → all digits significant → 4 sig figs

✔ 4

---

#### j) 0.0000455 → 455 → 3 sig figs

✔ 3

---

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

✔ 5

---

#### l) 30.0 × 10⁻² → 30.0 has 3 sig figs (trailing zero after decimal)

✔ 3

---

#### m) 0.982 × 10⁻³ → coefficient 0.982 → 982 → 3 sig figs

✔ 3

---

#### n) 0.0473 → 473 → 3 sig figs

✔ 3

---

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

✔ 6

---

#### p) 3.03 × 10⁻¹ → coefficient 3.03 → 3 sig figs

✔ 3

---

#### q) 20.4 × 10⁵ → 20.4 → 3 sig figs

✔ 3

---

#### r) 1.29 → 3 sig figs

✔ 3

---

#### s) 0.00565 → 565 → 3 sig figs

✔ 3

---

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

✔ 8

---

#### u) 450.0 × 10³ → 450.0 → 4 sig figs (trailing zero with decimal)

✔ 4

---

#### v) 1000 × 10⁻³ → 1000 → no decimal → ambiguous, but written as 1000 without decimal → usually considered 1 sig fig unless specified.

But here: 1000 × 10⁻³ = 1 → so 1.000? Wait — let's see:

1000 × 10⁻³ = 1 → which is 1, but original was 1000 (no decimal). So we assume 1 sig fig.

But if it were 1.000 × 10³, it would be 4. But here it's written as "1000", so 1 sig fig.

✔ 1

---

#### w) 546,000 ± 10 → uncertainty is ±10 → implies last two digits uncertain → so 546,000 → 3 sig figs (only 5,4,6 are certain)

✔ 3

---

#### x) 546,000 ± 1000 → uncertainty ±1000 → so last three digits uncertain → 546,000 → only 5 and 4 are reliable → 2 sig figs

✔ 2

---

✔ Summary for #2:

| Letter | Sig Figs |
|--------|----------|
| 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) | 2 |

---

#3. Convert each into decimal form

We’ll expand the powers of 10.

---

#### 1.56 × 10⁴
= 1.56 × 10,000 = 15,600

✔ 15,600

---

#### 0.56 × 10⁻²
= 0.56 × 0.01 = 0.0056

✔ 0.0056

---

#### 3.69 × 10⁻²
= 3.69 × 0.01 = 0.0369

✔ 0.0369

---

#### 736.9 × 10⁵
= 736.9 × 100,000 = 73,690,000

✔ 73,690,000

---

#### 0.00259 × 10⁵
= 0.00259 × 100,000 = 259

✔ 259

---

#### 0.000459 × 10⁻¹
= 0.000459 × 0.1 = 0.0000459

✔ 0.0000459

---

#### 13.69 × 10⁻²
= 13.69 × 0.01 = 0.1369

✔ 0.1369

---

#### 6.9 × 10⁴
= 6.9 × 10,000 = 69,000

✔ 69,000

---

#### 0.00259 × 10³
= 0.00259 × 1,000 = 2.59

✔ 2.59

---

#### 0.0209 × 10⁻³
= 0.0209 × 0.001 = 0.0000209

✔ 0.0000209

---

✔ Summary for #3:

| Expression | Decimal Form |
|----------------------|--------------------|
| 1.56 × 10⁴ | 15,600 |
| 0.56 × 10⁻² | 0.0056 |
| 3.69 × 10⁻² | 0.0369 |
| 736.9 × 10⁵ | 73,690,000 |
| 0.00259 × 10⁵ | 259 |
| 0.000459 × 10⁻¹ | 0.0000459 |
| 13.69 × 10⁻² | 0.1369 |
| 6.9 × 10⁴ | 69,000 |
| 0.00259 × 10³ | 2.59 |
| 0.0209 × 10⁻³ | 0.0000209 |

---

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

We must align exponents or use standard arithmetic.

---

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

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

✔ $2.653 \times 10^6$

---

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

$4.6 \times 10^3 = 4600$
$1913.0 - 4600 = -2687$

Now write in sci not: $-2.687 \times 10^3$

✔ $-2.687 \times 10^3$

---

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

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

✔ $2.532 \times 10^{24}$

---

#### d) $2.130 \times 10^3 - 6.6 \times 10^2$

$6.6 \times 10^2 = 0.66 \times 10^3$
$2.130 - 0.66 = 1.47$ → $1.47 \times 10^3$

✔ $1.47 \times 10^3$

---

#### e) $9.10 \times 10^3 + 2.2 \times 10^6$

Convert: $9.10 \times 10^3 = 0.00910 \times 10^6$
$0.00910 + 2.2 = 2.20910$ → $2.20910 \times 10^6$

✔ $2.2091 \times 10^6$ (round to 4 sig figs? 2.209 × 10⁶)

But 9.10 has 3 sig figs, 2.2 has 2 → so result should have 2 sig figs

Wait: 2.2 × 10⁶ has 2 sig figs, 9.10×10³ has 3 → when adding, precision matters.

2.2 × 10⁶ → precise to 10⁵ place
9.10 × 10³ → precise to 10¹ place → negligible

So sum ≈ 2.2 × 10⁶

✔ $2.2 \times 10^6$ (limited by 2.2)

---

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

$14.6 \times 10^2 = 1460$
$1113.0 - 1460 = -347$

→ $-3.47 \times 10^2$

✔ $-3.47 \times 10^2$

---

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

Convert: $6.18 \times 10^{-45} = 0.618 \times 10^{-44}$
Add: $0.618 + 4.72 = 5.338$ → $5.338 \times 10^{-44}$

✔ $5.338 \times 10^{-44}$

---

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

Convert: $4.25 \times 10^{-3} = 0.425 \times 10^{-2}$
$0.425 - 1.6 = -1.175$ → $-1.175 \times 10^{-2}$

Round to 2 sig figs? 1.6 has 2, 4.25 has 3 → subtraction → precision

1.6 × 10⁻² → ±0.1 × 10⁻² → so answer ≈ -1.2 × 10⁻²

✔ $-1.2 \times 10^{-2}$

---

✔ Summary for #4:

| Letter | Answer |
|--------|-----------------------------|
| a) | $2.653 \times 10^6$ |
| b) | $-2.687 \times 10^3$ |
| c) | $2.532 \times 10^{24}$ |
| d) | $1.47 \times 10^3$ |
| e) | $2.2 \times 10^6$ |
| f) | $-3.47 \times 10^2$ |
| g) | $5.338 \times 10^{-44}$ |
| h) | $-1.2 \times 10^{-2}$ |

---

#5. Calculate. Give answer in scientific notation

Use rules:
- Multiplication: add exponents
- Division: subtract exponents
- Keep correct sig figs

---

#### a) $3.95 \times 10^2 / 1.5 \times 10^6$

= $(3.95 / 1.5) \times 10^{2-6} = 2.633... \times 10^{-4}$

3.95 ÷ 1.5 = 2.633 → round to 2 sig figs (1.5 has 2) → 2.6 × 10⁻⁴

✔ $2.6 \times 10^{-4}$

---

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

= $3.5 \times 6.45 = 22.575$, $10^{2+10} = 10^{12}$

→ $2.2575 \times 10^{13}$

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

✔ $2.3 \times 10^{13}$

---

#### c) $4.44 \times 10^7 / 2.25 \times 10^5$

= $4.44 / 2.25 = 1.973...$, $10^{7-5} = 10^2$

→ $1.973 \times 10^2$

4.44 (3 sig), 2.25 (3 sig) → keep 3 → 1.97 × 10²

✔ $1.97 \times 10^2$

---

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

= $4.50 \times 3.67 = 16.515$, $10^{-12-12} = 10^{-24}$

→ $1.6515 \times 10^{-23}$

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

✔ $1.65 \times 10^{-23}$

---

#### e) $1.05 \times 10^{-26} / 4.2 \times 10^{56}$

= $1.05 / 4.2 = 0.25$, $10^{-26-56} = 10^{-82}$

→ $2.5 \times 10^{-83}$

1.05 (3 sig), 4.2 (2 sig) → round to 2 sig figs → 2.5 × 10⁻⁸³

✔ $2.5 \times 10^{-83}$

---

#### f) $(2.5 \times 10^9)(6.45 \times 10^4)$

= $2.5 \times 6.45 = 16.125$, $10^{9+4} = 10^{13}$

→ $1.6125 \times 10^{14}$

2.5 has 2 sig figs → 1.6 × 10¹⁴

✔ $1.6 \times 10^{14}$

---

#### g) $6.022 \times 10^{-23} / 3.011 \times 10^{-56}$

= $6.022 / 3.011 ≈ 2.000$, $10^{-23 - (-56)} = 10^{33}$

→ $2.000 \times 10^{33}$

Both numbers have 4 sig figs → keep 4 → 2.000 × 10³³

✔ $2.000 \times 10^{33}$

---

#### h) $(6.88 \times 10^2)(3.45 \times 10^{-10})$

= $6.88 \times 3.45 = 23.62$, $10^{2-10} = 10^{-8}$

→ $2.362 \times 10^{-7}$

6.88 (3 sig), 3.45 (3 sig) → 3 sig figs → 2.36 × 10⁻⁷

✔ $2.36 \times 10^{-7}$

---

✔ Summary for #5:

| Letter | Answer |
|--------|----------------------------|
| 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) | $2.000 \times 10^{33}$ |
| h) | $2.36 \times 10^{-7}$ |

---

✔ Final Answers Summary

If you need these in a printable format or want explanations on any specific rule, let me know! This covers all parts of the worksheet.