Sure, let's analyze each statement one by one:
### Statement 1: All perfect squares are even numbers.
- A perfect square is a number that can be expressed as [tex]\(n \times n\)[/tex] where [tex]\(n\)[/tex] is an integer.
- To determine whether all perfect squares are even, consider both odd and even integers.
- Example of an odd perfect square: [tex]\(1 = 1 \times 1\)[/tex], [tex]\(9 = 3 \times 3\)[/tex], [tex]\(25 = 5 \times 5\)[/tex].
- These examples show that odd integers can also produce perfect squares.
So, not all perfect squares are even numbers. The correct answer is:
False
### Statement 2: A negative number cannot be a perfect square.
- A perfect square is always the square of an integer.
- When you square any real number (positive or zero), the result is always non-negative (either positive or zero).
- Thus, there is no real integer whose square is negative.
So, a negative number cannot be a perfect square. The correct answer is:
True
### Statement 3: A perfect square cannot also be a perfect cube.
- A perfect square is a number that can be expressed as [tex]\(n^2\)[/tex] and a perfect cube is a number that can be expressed as [tex]\(n^3\)[/tex] where [tex]\(n\)[/tex] is an integer.
- Some numbers can satisfy both conditions. For example, [tex]\(64\)[/tex] is a perfect square ([tex]\(8^2\)[/tex]) and also a perfect cube ([tex]\(4^3\)[/tex]).
- Another example is [tex]\(1\)[/tex], which is both [tex]\(1^2\)[/tex] and [tex]\(1^3\)[/tex].
So, a number can be both a perfect square and a perfect cube. The correct answer is:
False
### Statement 4: The product of two equal integers is always a perfect square.
- If you take an integer [tex]\(n\)[/tex] and multiply it by itself ([tex]\(n \times n\)[/tex]), this can be written as [tex]\(n^2\)[/tex], which is the definition of a perfect square.
So, the product of two equal integers is always a perfect square. The correct answer is:
True
Thus, the answers are:
1. False
2. True
3. False
4. True
