Let's carefully examine why \(-5\) cannot be the smallest integer whose square lies between 15 and 30.

1. **Check the square of \(-5\):**
\[ (-5)^2 = 25 \]
Since \( 25 \) is indeed between 15 and 30, \(-5\) satisfies the condition for the square value.

2. **Check the requirement for "smallest integer":**
   - The term "smallest integer" typically refers to the integer with the smallest absolute value (the closest to zero) that meets the condition. 

3. **Considering negative values:**
   - Even though \(-5\) satisfies the condition for the square value, the smallest integer (considering absolute value and closeness to zero) whose square falls within the given range is \(4\).

So, while \(-5\) satisfies the square condition, the smallest positive integer whose square is between 15 and 30 is \(4\), because \(4\) is closer to zero than \(-5\).

Would you like more details or have any questions? Here are some additional questions you might find interesting:

1. What is the largest integer whose square is less than 100?
2. How do you find the square root of a number manually?
3. What is the smallest integer whose cube is greater than 100?
4. How can you approximate square roots without a calculator?
5. What is the difference between rational and irrational numbers?

Tip: When determining the "smallest" or "largest" integer for mathematical problems, always consider the absolute value and the proximity to zero unless specifically asked for negative values.

Explore why -5 cannot be considered the smallest integer whose square lies between 15 and 30. Learn about integer analysis and absolute value considerations in mathematical problem-solving. Suitable for students in Grades 6-8.

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