Scientific Notation Practice Worksheet with Examples and Problems

Problem: Simplify and express each problem in scientific notation.

We will solve each problem step by step, following the principles of scientific notation. Scientific notation is expressed as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

---

#### Example 1:
\[
(2 \times 10^4) + (3 \times 10^5)
\]
- Convert both terms to have the same power of 10:
\[
(2 \times 10^4) + (3 \times 10^5) = (2 \times 10^4) + (3 \times 10^4 \times 10) = (2 \times 10^4) + (30 \times 10^4)
\]
- Add the coefficients:
\[
2 + 30 = 32
\]
- Combine:
\[
32 \times 10^4
\]
- Adjust to scientific notation (\( 1 \leq a < 10 \)):
\[
32 \times 10^4 = 3.2 \times 10^5
\]

Answer:
\[
\boxed{3.2 \times 10^5}
\]

---

#### Example 2:
\[
(7 \times 10^8) - (4 \times 10^6)
\]
- Convert both terms to have the same power of 10:
\[
(7 \times 10^8) - (4 \times 10^6) = (7 \times 10^6 \times 10^2) - (4 \times 10^6) = (700 \times 10^6) - (4 \times 10^6)
\]
- Subtract the coefficients:
\[
700 - 4 = 696
\]
- Combine:
\[
696 \times 10^6
\]
- Adjust to scientific notation (\( 1 \leq a < 10 \)):
\[
696 \times 10^6 = 6.96 \times 10^8
\]

Answer:
\[
\boxed{6.96 \times 10^8}
\]

---

Problems to Solve:

#### Problem 1:
\[
(2 \times 10^3) + (5 \times 10^5)
\]
- Convert both terms to have the same power of 10:
\[
(2 \times 10^3) + (5 \times 10^5) = (2 \times 10^3) + (5 \times 10^3 \times 10^2) = (2 \times 10^3) + (500 \times 10^3)
\]
- Add the coefficients:
\[
2 + 500 = 502
\]
- Combine:
\[
502 \times 10^3
\]
- Adjust to scientific notation:
\[
502 \times 10^3 = 5.02 \times 10^5
\]

Answer:
\[
\boxed{5.02 \times 10^5}
\]

#### Problem 2:
\[
(4 \times 10^8) - (9 \times 10^7)
\]
- Convert both terms to have the same power of 10:
\[
(4 \times 10^8) - (9 \times 10^7) = (4 \times 10^7 \times 10) - (9 \times 10^7) = (40 \times 10^7) - (9 \times 10^7)
\]
- Subtract the coefficients:
\[
40 - 9 = 31
\]
- Combine:
\[
31 \times 10^7
\]
- Adjust to scientific notation:
\[
31 \times 10^7 = 3.1 \times 10^8
\]

Answer:
\[
\boxed{3.1 \times 10^8}
\]

#### Problem 3:
\[
(8 \times 10^9) - (3 \times 10^7)
\]
- Convert both terms to have the same power of 10:
\[
(8 \times 10^9) - (3 \times 10^7) = (8 \times 10^7 \times 10^2) - (3 \times 10^7) = (800 \times 10^7) - (3 \times 10^7)
\]
- Subtract the coefficients:
\[
800 - 3 = 797
\]
- Combine:
\[
797 \times 10^7
\]
- Adjust to scientific notation:
\[
797 \times 10^7 = 7.97 \times 10^9
\]

Answer:
\[
\boxed{7.97 \times 10^9}
\]

#### Problem 4:
\[
(5 \times 10^7) + (1 \times 10^6)
\]
- Convert both terms to have the same power of 10:
\[
(5 \times 10^7) + (1 \times 10^6) = (5 \times 10^6 \times 10) + (1 \times 10^6) = (50 \times 10^6) + (1 \times 10^6)
\]
- Add the coefficients:
\[
50 + 1 = 51
\]
- Combine:
\[
51 \times 10^6
\]
- Adjust to scientific notation:
\[
51 \times 10^6 = 5.1 \times 10^7
\]

Answer:
\[
\boxed{5.1 \times 10^7}
\]

#### Problem 5:
\[
(4 \times 10^2) + (7 \times 10^3)
\]
- Convert both terms to have the same power of 10:
\[
(4 \times 10^2) + (7 \times 10^3) = (4 \times 10^2) + (7 \times 10^2 \times 10) = (4 \times 10^2) + (70 \times 10^2)
\]
- Add the coefficients:
\[
4 + 70 = 74
\]
- Combine:
\[
74 \times 10^2
\]
- Adjust to scientific notation:
\[
74 \times 10^2 = 7.4 \times 10^3
\]

Answer:
\[
\boxed{7.4 \times 10^3}
\]

#### Problem 6:
\[
(9 \times 10^5) - (6 \times 10^4)
\]
- Convert both terms to have the same power of 10:
\[
(9 \times 10^5) - (6 \times 10^4) = (9 \times 10^4 \times 10) - (6 \times 10^4) = (90 \times 10^4) - (6 \times 10^4)
\]
- Subtract the coefficients:
\[
90 - 6 = 84
\]
- Combine:
\[
84 \times 10^4
\]
- Adjust to scientific notation:
\[
84 \times 10^4 = 8.4 \times 10^5
\]

Answer:
\[
\boxed{8.4 \times 10^5}
\]

#### Problem 7:
\[
(1 \times 10^6) - (8 \times 10^4)
\]
- Convert both terms to have the same power of 10:
\[
(1 \times 10^6) - (8 \times 10^4) = (1 \times 10^4 \times 10^2) - (8 \times 10^4) = (100 \times 10^4) - (8 \times 10^4)
\]
- Subtract the coefficients:
\[
100 - 8 = 92
\]
- Combine:
\[
92 \times 10^4
\]
- Adjust to scientific notation:
\[
92 \times 10^4 = 9.2 \times 10^5
\]

Answer:
\[
\boxed{9.2 \times 10^5}
\]

#### Problem 8:
\[
(5 \times 10^9) + (3 \times 10^8)
\]
- Convert both terms to have the same power of 10:
\[
(5 \times 10^9) + (3 \times 10^8) = (5 \times 10^8 \times 10) + (3 \times 10^8) = (50 \times 10^8) + (3 \times 10^8)
\]
- Add the coefficients:
\[
50 + 3 = 53
\]
- Combine:
\[
53 \times 10^8
\]
- Adjust to scientific notation:
\[
53 \times 10^8 = 5.3 \times 10^9
\]

Answer:
\[
\boxed{5.3 \times 10^9}
\]

---

Final Answers:

1. \(\boxed{5.02 \times 10^5}\)
2. \(\boxed{3.1 \times 10^8}\)
3. \(\boxed{7.97 \times 10^9}\)
4. \(\boxed{5.1 \times 10^7}\)
5. \(\boxed{7.4 \times 10^3}\)
6. \(\boxed{8.4 \times 10^5}\)
7. \(\boxed{9.2 \times 10^5}\)
8. \(\boxed{5.3 \times 10^9}\)