To solve the problem of finding the smallest number that must be subtracted from 3400 to make it a perfect cube, let's go through a detailed step-by-step solution.
1. Identify the given number: We are given the number 3400.
2. Find the cube root of the given number: We need to determine the cube root of 3400 and round it down to the nearest integer. This gives us an approximate idea of where the closest perfect cube lies. For 3400, the nearest integer cube root is 15.
3. Calculate the nearest perfect cube: Next, we compute the cube of the nearest integer found in the previous step.
[tex]\[
15^3 = 3375
\][/tex]
4. Determine the difference: Finally, we subtract this perfect cube from the original number to find the smallest number that must be subtracted from 3400.
[tex]\[
3400 - 3375 = 25
\][/tex]
So, the smallest number that must be subtracted from 3400 to make it a perfect cube is 25. Therefore, the correct answer is:
[tex]\[
\boxed{25}
\][/tex]
