Question
Which of these are square numbers? \( \square 36 \) \( \square 5 \) \( \square 4^{2} \) \( \square 3^{2} \times 6^{2} \) \( \square 10 \) \( \square 5^{8} \)
Ask by Hall Mcfarlane. in Oman
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The square numbers are 36, \(4^2\), \(3^2 \times 6^2\), and \(5^8\).
Solution
Let's evaluate each of the given expressions to determine whether they are perfect square numbers.
1. **36**
- **Explanation:** \( 6^2 = 36 \)
- **Conclusion:** **Square Number**
2. **5**
- **Explanation:** 5 is not a perfect square since there's no integer \( n \) such that \( n^2 = 5 \).
- **Conclusion:** **Not a Square Number**
3. **\( 4^2 \)**
- **Explanation:** By definition, \( 4^2 = 16 \), which is a perfect square.
- **Conclusion:** **Square Number**
4. **\( 3^2 \times 6^2 \)**
- **Simplification:** \( 3^2 \times 6^2 = (3 \times 6)^2 = 18^2 = 324 \)
- **Conclusion:** **Square Number**
5. **10**
- **Explanation:** 10 is not a perfect square since there's no integer \( n \) such that \( n^2 = 10 \).
- **Conclusion:** **Not a Square Number**
6. **\( 5^8 \)**
- **Simplification:** \( 5^8 = (5^4)^2 = 625^2 \)
- **Conclusion:** **Square Number**
### **Summary of Square Numbers:**
- **36**
- **\( 4^2 \)**
- **\( 3^2 \times 6^2 \)**
- **\( 5^8 \)**
These expressions represent perfect square numbers.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
A square number is an integer that is the square of an integer. For example, \(36\) is a square number because it can be expressed as \(6^2\). Thus, both \(4^{2}\) (which equals \(16\)) and \(3^{2} \times 6^{2}\) (which simplifies to \(9 \times 36 = 324 = 18^2\)) also result in square numbers as they're formed by squaring integers. On the flip side, \(5\) and \(10\) are not square numbers, as there are no integers whose square gives those values. \(5^{8}\) is \(390625\), which is also a square, as it equals \(5^4 \times 5^4\) or \((5^4)^2\). So the numbers in the list that are square numbers include \(36\), \(4^{2}\), \(3^{2} \times 6^{2}\), and \(5^{8}\).
