Is 512 a perfect cube? If not, by which smallest natural number should it be multiplied to make it a perfect cube?

To determine if 512 is a perfect cube and, if not, by which smallest natural number it should be multiplied to become a perfect cube, we can follow these steps:

1. Break Down the Problem

   • Check if 512 is a perfect cube by finding its prime factorization.
   • Determine if all prime factors appear in multiples of three.
   • If not, find the smallest number to multiply with 512 to make all prime factors appear in multiples of three.

2. Relevant Concepts

   • A perfect cube is a number that can be expressed as $n^3$, where $n$ is an integer.
   • Prime factorization helps in determining the powers of prime factors.

3. Analysis and Detail

   • Prime Factorization of 512: $512 \div 2 = 256 \\ 256 \div 2 = 128 \\ 128 \div 2 = 64 \\ 64 \div 2 = 32 \\ 32 \div 2 = 16 \\ 16 \div 2 = 8 \\ 8 \div 2 = 4 \\ 4 \div 2 = 2 \\ 2 \div 2 = 1$ Thus, the prime factorization of 512 is $2^9$.

   • Checking for Perfect Cube:

     • For 512 to be a perfect cube, the power of each prime factor must be a multiple of 3.
     • Here, the power of 2 is 9, which is already a multiple of 3. Therefore, 512 is a perfect cube.

4. Verify and Summarize

   • Since the power of 2 in the prime factorization of 512 is 9, which is a multiple of 3, 512 is indeed a perfect cube.

Final Answer

512 is a perfect cube. There is no need to multiply it by any number to make it a perfect cube.