Which statement about perfect cubes is true?
A. 25 is a perfect cube because [tex]$25=5+5+5+5+5$[/tex]
B. 30 is a perfect cube because [tex]$30=3 \bullet 10$[/tex]
C. 512 is a perfect cube because [tex]$512=8 \cdot 8 \cdot 8$[/tex]
D. 1,875 is a perfect cube because [tex]$1,875=25 \bullet 25 \bullet 3$[/tex]
Answer (2)
A perfect cube is a number that can be expressed as n 3 for some integer n .
Check each statement to see if the given number is a perfect cube.
25 is not a perfect cube.
30 is not a perfect cube.
512 is a perfect cube since 512 = 8 3 .
1,875 is not a perfect cube.
The correct statement is that 512 is a perfect cube because 512 = 8 ⋅ 8 ⋅ 8 = 8 3 . Thus, the answer is 512 .
Explanation
Understanding Perfect Cubes We are given four statements about perfect cubes and we need to determine which one is true. A perfect cube is a number that can be obtained by cubing an integer. In other words, a number x is a perfect cube if there exists an integer n such that x = n 3 .
Analyzing Statement 1 Statement 1: 25 is a perfect cube because 25 = 5 + 5 + 5 + 5 + 5 . This statement is incorrect because 25 is not a perfect cube. The cube root of 25 is approximately 2.92, which is not an integer. Also, the reasoning provided ( 25 = 5 + 5 + 5 + 5 + 5 ) is irrelevant to determining if a number is a perfect cube.
Analyzing Statement 2 Statement 2: 30 is a perfect cube because 30 = 3 ⋅ 10 . This statement is also incorrect. The cube root of 30 is approximately 3.11, which is not an integer. The reasoning ( 30 = 3 ⋅ 10 ) is irrelevant.
Analyzing Statement 3 Statement 3: 512 is a perfect cube because 512 = 8 ⋅ 8 ⋅ 8 . This statement is correct because 512 = 8 3 . Thus, 512 is a perfect cube.
Analyzing Statement 4 Statement 4: 1,875 is a perfect cube because 1 , 875 = 25 ⋅ 25 ⋅ 3 . This statement is incorrect because 1875 is not a perfect cube. The cube root of 1875 is approximately 12.33, which is not an integer.
Conclusion Therefore, the true statement is that 512 is a perfect cube because 512 = 8 ⋅ 8 ⋅ 8 .
Examples
Perfect cubes are useful in various real-world applications, such as calculating the volume of a cube. For example, if you have a cubic box with sides of equal length, the volume of the box is a perfect cube. If the side length of the box is 5 units, then the volume is 5 3 = 125 cubic units. Understanding perfect cubes helps in determining the dimensions or volume of such objects.
The only true statement about perfect cubes is that 512 is a perfect cube because it can be expressed as 8 3 . Other statements about 25, 30, and 1,875 being perfect cubes are incorrect. Therefore, the correct choice is C.
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