Which statements are true? Select all that apply.
81 is a perfect square.
75 is a perfect cube.
729 is a perfect cube.
32 is neither a perfect square nor a perfect cube.
81 is both a perfect square and a perfect cube.
The Correct Answer and Explanation is :
Correct Answers:
- 81 is a perfect square. ✅
- 729 is a perfect cube. ✅
- 32 is neither a perfect square nor a perfect cube. ✅
Explanation:
1. 81 is a perfect square.
- A perfect square is a number that can be expressed as the square of an integer.
[
9 \times 9 = 81
]
Since 81 is ( 9^2 ), it is a perfect square.
2. 75 is a perfect cube.
- A perfect cube is a number that can be expressed as the cube of an integer.
Testing integers:
[
4^3 = 64, \quad 5^3 = 125
]
Since 75 is not between ( 4^3 ) and ( 5^3 ), it is not a perfect cube.
This statement is false.
3. 729 is a perfect cube.
- ( 729 = 9 \times 9 \times 9 = 9^3 ).
Since it can be expressed as the cube of an integer (( 9 )), it is a perfect cube.
4. 32 is neither a perfect square nor a perfect cube.
- Testing for perfect squares:
[
5^2 = 25, \quad 6^2 = 36
]
Since ( 32 ) is not the square of any integer, it is not a perfect square. - Testing for perfect cubes:
[
3^3 = 27, \quad 4^3 = 64
]
Similarly, ( 32 ) is not the cube of any integer, so it is not a perfect cube.
This statement is true.
5. 81 is both a perfect square and a perfect cube.
- ( 81 = 9^2 ), so it is a perfect square.
- Testing for cubes:
[
4^3 = 64, \quad 5^3 = 125
]
( 81 ) is not a perfect cube.
This statement is false.
Final Answer:
- True Statements:
- 81 is a perfect square.
- 729 is a perfect cube.
- 32 is neither a perfect square nor a perfect cube.