5. Operations with Radical Expressions
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You need to find three positive whole numbers a, b, and c such that a2=b3=c4.
Example Solution: 4096
Since 16 cannot be written as a whole number cube, c cannot be 2. Let's make a table where we try some more numbers.
c | c4=k | a2=k | b3=k |
---|---|---|---|
3 | 34=81 | 92=81 | No way |
4 | 44=256 | 162=256 | No way |
5 | 54=625 | 252=625 | No way |
6 | 64=1296 | 362=1296 | No way |
7 | 74=2041 | 492=2041 | No way |
8 | 84=4096 | 642=4096 | 163=4096 |
From the table above, one number that satisfies the required condition is 4096.
k | k | 3k | 4k |
---|---|---|---|
212=4096 | 4096=64 | 34096=16 | 44096=8 |
312=531441 | 531441=729 | 3531441=81 | 4531441=27 |
412=16777216 | 16777216=4096 | 316777216=256 | 416777216=64 |
Therefore, any number raised to the 12th power will satisfy the given condition. Particularly, we can pick 4096.