Let's solve the problem step-by-step:
1. Check if 576 is a perfect cube:
- To determine if a number is a perfect cube, we can take its cube root and see if the resulting value is an integer.
- The cube root of 576 is approximately 8.329. Since 8.329 is not an integer, 576 is not a perfect cube.
2. Find the smallest number to multiply 576 to make it a perfect cube:
- To transform 576 into a perfect cube, we need to adjust its prime factors so that each prime factor's exponent is divisible by 3.
- The prime factorization of 576 is:
[tex]\[
576 = 2^6 \times 3^2
\][/tex]
- To make the exponents of these prime factors divisible by 3:
- For [tex]\(2^6\)[/tex]: We need the exponent to be a multiple of 3. The closest multiple of 3 greater than 6 is 6 itself, which is already perfect.
- For [tex]\(3^2\)[/tex]: The closest multiple of 3 greater than 2 is 3. This means we need one more factor of 3 to make the exponent a multiple of 3.
- Therefore, we need to multiply 576 by 3 to make the prime factor exponents all divisible by 3.
3. New number and verification:
- Multiplying 576 by 3 gives:
[tex]\[
576 \times 3 = 1728
\][/tex]
- Now, we can check if 1728 is a perfect cube:
[tex]\[
\sqrt[3]{1728} = 12
\][/tex]
Since 12 is an integer, 1728 is a perfect cube.
Conclusion:
576 is not a perfect cube. The smallest number that must be multiplied to 576 to make it a perfect cube is 3, resulting in 1728, which has a cube root of 12.
