in which of the system 384 is equal to 1234? 1.base 5 2.base 6 3.base 7?

Answer 1

None of the above:

1234 base 7 is 466
1 x 7^3 = 1 x 343 = 343
2 x 7^2 = 2 x 49 = 98
3 x 7^1 = 3 x 7 = 21
4 x 7^0 = 4 x 1 = 4

1234 base 6 is 310
1 x 6^3 = 1 x 216 = 216
2 x 6^2 = 2 x 36 = 72
3 x 6^1 = 3 x 6 = 18
4 x 6^0 = 4 x 1 = 4

1234 base 5 is 194
1 x 5^3 = 1 x 125 = 125
2 x 5^2 = 2 x 25 = 50
3 x 5^1 = 3 x 5 = 15
4 x 5^0 = 4 x 1 = 4

Working the other direction, 384 is:
3014 base 5
1440 base 6
1056 base 7

So the correct answer is 4. none of the above.

Answer 2

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Answer 3

If the base is b, then you have:

b^3 + 2 b^2 + 3 b + 4 = 384

or

b^3 + 2 b^2 + 3 b - 380 = 0

According to the online cubic equation solver at

http://www.1728.com/cubic.htm

the numerical value of the three roots are approximately

6.508817513455076
-4.254408756727538 + i* 6.346837525970616
-4.254408756727538 - i* 6.346837525970616

where the solution with zero imaginary part is a little over halfway from base six to base seven.