Patterns in Numbers - Using Exponents Printable (5th - 6th Grade ... - Free Printable

Problem Explanation and Solution

The worksheet focuses on understanding exponents, writing numbers in exponential form, expanded form, standard form, and comparing exponential expressions. Let's solve each section step by step.

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#### Section 1: Write using exponents

1. 3 × 3 × 3 × 3
- This is \(3\) multiplied by itself 4 times.
- Exponential form: \(3^4\)

2. 364 × 364
- This is \(364\) multiplied by itself 2 times.
- Exponential form: \(364^2\)

3. 2 × 2 × 2 × 2 × 2 × 2 × 2
- This is \(2\) multiplied by itself 7 times.
- Exponential form: \(2^7\)

4. 13 × 13 × 13
- This is \(13\) multiplied by itself 3 times.
- Exponential form: \(13^3\)

5. 8 × 8 × 8 × 7 × 7
- This can be written as \(8^3 \times 7^2\).
- Exponential form: \(8^3 \times 7^2\)

6. 49
- Recognize that \(49 = 7 \times 7\).
- Exponential form: \(7^2\)

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#### Section 2: Write in expanded form

7. \(10^4\)
- Expanded form: \(10 \times 10 \times 10 \times 10\)

8. \(6^5\)
- Expanded form: \(6 \times 6 \times 6 \times 6 \times 6\)

9. \(3^2\)
- Expanded form: \(3 \times 3\)

10. \(7^3\)
- Expanded form: \(7 \times 7 \times 7\)

11. \(12^4\)
- Expanded form: \(12 \times 12 \times 12 \times 12\)

12. 5 cubed
- "Cubed" means raised to the power of 3.
- Expanded form: \(5 \times 5 \times 5\)

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#### Section 3: Write in standard form

13. \(5^4\)
- Calculate: \(5^4 = 5 \times 5 \times 5 \times 5 = 625\)
- Standard form: \(625\)

14. \(2^6\)
- Calculate: \(2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\)
- Standard form: \(64\)

15. 11 squared
- "Squared" means raised to the power of 2.
- Calculate: \(11^2 = 11 \times 11 = 121\)
- Standard form: \(121\)

16. \(10^7\)
- Calculate: \(10^7 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10,000,000\)
- Standard form: \(10,000,000\)

17. \(12^2\)
- Calculate: \(12^2 = 12 \times 12 = 144\)
- Standard form: \(144\)

18. 6 cubed
- "Cubed" means raised to the power of 3.
- Calculate: \(6^3 = 6 \times 6 \times 6 = 216\)
- Standard form: \(216\)

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#### Section 4: Compare using <, >, or =

19. \(4^2 \quad \square \quad 2^4\)
- Calculate \(4^2 = 4 \times 4 = 16\)
- Calculate \(2^4 = 2 \times 2 \times 2 \times 2 = 16\)
- Comparison: \(4^2 = 2^4\)

20. \(4^3 \quad \square \quad 3^4\)
- Calculate \(4^3 = 4 \times 4 \times 4 = 64\)
- Calculate \(3^4 = 3 \times 3 \times 3 \times 3 = 81\)
- Comparison: \(4^3 < 3^4\)

21. \(5^8 \quad \square \quad 5^9\)
- Since the base is the same (\(5\)), compare the exponents.
- \(8 < 9\), so \(5^8 < 5^9\)
- Comparison: \(5^8 < 5^9\)

22. \(3^8 \quad \square \quad 3 \times 8\)
- Calculate \(3^8 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561\)
- Calculate \(3 \times 8 = 24\)
- Comparison: \(3^8 > 3 \times 8\)

23. \(2^5 \quad \square \quad 5^2\)
- Calculate \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\)
- Calculate \(5^2 = 5 \times 5 = 25\)
- Comparison: \(2^5 > 5^2\)

24. \(10^3 \quad \square \quad 10 + 10 + 10\)
- Calculate \(10^3 = 10 \times 10 \times 10 = 1000\)
- Calculate \(10 + 10 + 10 = 30\)
- Comparison: \(10^3 > 10 + 10 + 10\)

25. \(5^3 \quad \square \quad 5 \times 5 \times 5\)
- Calculate \(5^3 = 5 \times 5 \times 5 = 125\)
- The right side is already \(5 \times 5 \times 5 = 125\)
- Comparison: \(5^3 = 5 \times 5 \times 5\)

26. \(7^3 \quad \square \quad 3^7\)
- Calculate \(7^3 = 7 \times 7 \times 7 = 343\)
- Calculate \(3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187\)
- Comparison: \(7^3 < 3^7\)

27. \(10^4 \quad \square \quad 4 \times 10\)
- Calculate \(10^4 = 10 \times 10 \times 10 \times 10 = 10,000\)
- Calculate \(4 \times 10 = 40\)
- Comparison: \(10^4 > 4 \times 10\)

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Final Answers

1. \(3^4\)
2. \(364^2\)
3. \(2^7\)
4. \(13^3\)
5. \(8^3 \times 7^2\)
6. \(7^2\)

7. \(10 \times 10 \times 10 \times 10\)
8. \(6 \times 6 \times 6 \times 6 \times 6\)
9. \(3 \times 3\)
10. \(7 \times 7 \times 7\)
11. \(12 \times 12 \times 12 \times 12\)
12. \(5 \times 5 \times 5\)

13. \(625\)
14. \(64\)
15. \(121\)
16. \(10,000,000\)
17. \(144\)
18. \(216\)

19. \(=\)
20. \(<\)
21. \(<\)
22. \(>\)
23. \(>\)
24. \(>\)
25. \(=\)
26. \(<\)
27. \(>\)

Final Answer Boxed:
\[
\boxed{
\begin{array}{l}
1. 3^4 \\
2. 364^2 \\
3. 2^7 \\
4. 13^3 \\
5. 8^3 \times 7^2 \\
6. 7^2 \\
7. 10 \times 10 \times 10 \times 10 \\
8. 6 \times 6 \times 6 \times 6 \times 6 \\
9. 3 \times 3 \\
10. 7 \times 7 \times 7 \\
11. 12 \times 12 \times 12 \times 12 \\
12. 5 \times 5 \times 5 \\
13. 625 \\
14. 64 \\
15. 121 \\
16. 10,000,000 \\
17. 144 \\
18. 216 \\
19. = \\
20. < \\
21. < \\
22. > \\
23. > \\
24. > \\
25. = \\
26. < \\
27. >
\end{array}
}
\]