Is 864 a perfect cube? This question often arises when dealing with number properties and algebraic expressions. To answer this, we need to understand what a perfect cube is and how to determine if a given number is a perfect cube.
A perfect cube is a number that can be expressed as the cube of an integer. In other words, if we have a number ‘n’ and we can find an integer ‘a’ such that \( a^3 = n \), then ‘n’ is a perfect cube. For example, 27 is a perfect cube because \( 3^3 = 27 \), while 28 is not a perfect cube because there is no integer ‘a’ such that \( a^3 = 28 \).
Now, let’s examine the number 864. To determine if it is a perfect cube, we need to find an integer ‘a’ such that \( a^3 = 864 \). One way to do this is by using prime factorization. By breaking down 864 into its prime factors, we can then determine if the exponents of these factors are all multiples of 3, which would indicate that 864 is a perfect cube.
The prime factorization of 864 is as follows:
\( 864 = 2^5 \times 3^3 \)
Here, we have two prime factors: 2 and 3. The exponent of 2 is 5, which is not a multiple of 3, and the exponent of 3 is 3, which is a multiple of 3. Since the exponent of 2 is not a multiple of 3, we can conclude that 864 is not a perfect cube.
In conclusion, the answer to the question “Is 864 a perfect cube?” is no. While 864 has a prime factorization that includes a cube of 3, the exponent of the other prime factor, 2, is not a multiple of 3, which means that 864 cannot be expressed as the cube of an integer.
