Worksheet 58: Operations with Scientific Notation, designed for chemistry students to practice mathematical operations using scientific notation.

Problem: Operations with Scientific Notation

The worksheet asks us to perform arithmetic operations (addition, subtraction, multiplication, and division) on numbers expressed in scientific notation and express the answers in scientific notation. Let's solve each part step by step.

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Part 1: Perform the following operations and express the answers in scientific notation.

#### a. \( (1.2 \times 10^5) + (5.35 \times 10^6) \)

1. Align the powers of 10: To add these numbers, we need to express them with the same power of 10.
- \( 1.2 \times 10^5 = 0.12 \times 10^6 \)
- So, the expression becomes:
\[
(0.12 \times 10^6) + (5.35 \times 10^6)
\]

2. Add the coefficients:
\[
0.12 + 5.35 = 5.47
\]

3. Combine with the common power of 10:
\[
5.47 \times 10^6
\]

Answer:
\[
\boxed{5.47 \times 10^6}
\]

---

#### b. \( (6.91 \times 10^{-2}) + (2.4 \times 10^{-3}) \)

1. Align the powers of 10:
- \( 6.91 \times 10^{-2} = 69.1 \times 10^{-3} \)
- So, the expression becomes:
\[
(69.1 \times 10^{-3}) + (2.4 \times 10^{-3})
\]

2. Add the coefficients:
\[
69.1 + 2.4 = 71.5
\]

3. Combine with the common power of 10:
\[
71.5 \times 10^{-3}
\]

4. Convert to proper scientific notation:
- Move the decimal point one place to the left to get a coefficient between 1 and 10:
\[
71.5 \times 10^{-3} = 7.15 \times 10^{-2}
\]

Answer:
\[
\boxed{7.15 \times 10^{-2}}
\]

---

#### c. \( (9.70 \times 10^6) + (8.3 \times 10^5) \)

1. Align the powers of 10:
- \( 8.3 \times 10^5 = 0.83 \times 10^6 \)
- So, the expression becomes:
\[
(9.70 \times 10^6) + (0.83 \times 10^6)
\]

2. Add the coefficients:
\[
9.70 + 0.83 = 10.53
\]

3. Combine with the common power of 10:
\[
10.53 \times 10^6
\]

4. Convert to proper scientific notation:
- Move the decimal point one place to the left to get a coefficient between 1 and 10:
\[
10.53 \times 10^6 = 1.053 \times 10^7
\]

Answer:
\[
\boxed{1.053 \times 10^7}
\]

---

#### d. \( (3.67 \times 10^2) - (1.6 \times 10^1) \)

1. Align the powers of 10:
- \( 1.6 \times 10^1 = 0.16 \times 10^2 \)
- So, the expression becomes:
\[
(3.67 \times 10^2) - (0.16 \times 10^2)
\]

2. Subtract the coefficients:
\[
3.67 - 0.16 = 3.51
\]

3. Combine with the common power of 10:
\[
3.51 \times 10^2
\]

Answer:
\[
\boxed{3.51 \times 10^2}
\]

---

#### e. \( (8.41 \times 10^{-5}) - (7.9 \times 10^{-6}) \)

1. Align the powers of 10:
- \( 7.9 \times 10^{-6} = 0.79 \times 10^{-5} \)
- So, the expression becomes:
\[
(8.41 \times 10^{-5}) - (0.79 \times 10^{-5})
\]

2. Subtract the coefficients:
\[
8.41 - 0.79 = 7.62
\]

3. Combine with the common power of 10:
\[
7.62 \times 10^{-5}
\]

Answer:
\[
\boxed{7.62 \times 10^{-5}}
\]

---

#### f. \( (1.33 \times 10^5) - (4.9 \times 10^4) \)

1. Align the powers of 10:
- \( 4.9 \times 10^4 = 0.49 \times 10^5 \)
- So, the expression becomes:
\[
(1.33 \times 10^5) - (0.49 \times 10^5)
\]

2. Subtract the coefficients:
\[
1.33 - 0.49 = 0.84
\]

3. Combine with the common power of 10:
\[
0.84 \times 10^5
\]

4. Convert to proper scientific notation:
- Move the decimal point one place to the right to get a coefficient between 1 and 10:
\[
0.84 \times 10^5 = 8.4 \times 10^4
\]

Answer:
\[
\boxed{8.4 \times 10^4}
\]

---

Part 2: Perform the following operations and express the answers in scientific notation.

#### a. \( (4.3 \times 10^8) \times (2.0 \times 10^6) \)

1. Multiply the coefficients:
\[
4.3 \times 2.0 = 8.6
\]

2. Add the exponents:
\[
10^8 \times 10^6 = 10^{8+6} = 10^{14}
\]

3. Combine:
\[
8.6 \times 10^{14}
\]

Answer:
\[
\boxed{8.6 \times 10^{14}}
\]

---

#### b. \( (6.0 \times 10^3) \times (1.5 \times 10^{-2}) \)

1. Multiply the coefficients:
\[
6.0 \times 1.5 = 9.0
\]

2. Add the exponents:
\[
10^3 \times 10^{-2} = 10^{3-2} = 10^1
\]

3. Combine:
\[
9.0 \times 10^1
\]

Answer:
\[
\boxed{9.0 \times 10^1}
\]

---

#### c. \( (1.5 \times 10^{-2}) \times (8.0 \times 10^{-1}) \)

1. Multiply the coefficients:
\[
1.5 \times 8.0 = 12.0
\]

2. Add the exponents:
\[
10^{-2} \times 10^{-1} = 10^{-2-1} = 10^{-3}
\]

3. Combine:
\[
12.0 \times 10^{-3}
\]

4. Convert to proper scientific notation:
- Move the decimal point one place to the left to get a coefficient between 1 and 10:
\[
12.0 \times 10^{-3} = 1.2 \times 10^{-2}
\]

Answer:
\[
\boxed{1.2 \times 10^{-2}}
\]

---

#### d. \( \frac{7.8 \times 10^3}{1.2 \times 10^4} \)

1. Divide the coefficients:
\[
\frac{7.8}{1.2} = 6.5
\]

2. Subtract the exponents:
\[
\frac{10^3}{10^4} = 10^{3-4} = 10^{-1}
\]

3. Combine:
\[
6.5 \times 10^{-1}
\]

Answer:
\[
\boxed{6.5 \times 10^{-1}}
\]

---

#### e. \( \frac{8.1 \times 10^{-2}}{9.0 \times 10^2} \)

1. Divide the coefficients:
\[
\frac{8.1}{9.0} = 0.9
\]

2. Subtract the exponents:
\[
\frac{10^{-2}}{10^2} = 10^{-2-2} = 10^{-4}
\]

3. Combine:
\[
0.9 \times 10^{-4}
\]

4. Convert to proper scientific notation:
- Move the decimal point one place to the right to get a coefficient between 1 and 10:
\[
0.9 \times 10^{-4} = 9.0 \times 10^{-5}
\]

Answer:
\[
\boxed{9.0 \times 10^{-5}}
\]

---

#### f. \( \frac{6.48 \times 10^5}{(2.4 \times 10^4)(1.8 \times 10^{-2})} \)

1. Simplify the denominator:
- Multiply the coefficients:
\[
2.4 \times 1.8 = 4.32
\]
- Add the exponents:
\[
10^4 \times 10^{-2} = 10^{4-2} = 10^2
\]
- Combine:
\[
4.32 \times 10^2
\]

2. Rewrite the expression:
\[
\frac{6.48 \times 10^5}{4.32 \times 10^2}
\]

3. Divide the coefficients:
\[
\frac{6.48}{4.32} = 1.5
\]

4. Subtract the exponents:
\[
\frac{10^5}{10^2} = 10^{5-2} = 10^3
\]

5. Combine:
\[
1.5 \times 10^3
\]

Answer:
\[
\boxed{1.5 \times 10^3}
\]

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Final Answers:

1. a. \( \boxed{5.47 \times 10^6} \)
b. \( \boxed{7.15 \times 10^{-2}} \)
c. \( \boxed{1.053 \times 10^7} \)
d. \( \boxed{3.51 \times 10^2} \)
e. \( \boxed{7.62 \times 10^{-5}} \)
f. \( \boxed{8.4 \times 10^4} \)

2. a. \( \boxed{8.6 \times 10^{14}} \)
b. \( \boxed{9.0 \times 10^1} \)
c. \( \boxed{1.2 \times 10^{-2}} \)
d. \( \boxed{6.5 \times 10^{-1}} \)
e. \( \boxed{9.0 \times 10^{-5}} \)
f. \( \boxed{1.5 \times 10^3} \)