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24.
First, find the prime factorization of 13824:
13824 = 29 x 33.
When you take the cube root of this number, every grouping of 3 of the same factors will, upon "cube-rooting", become just 1 of that factor. So the first group of three 2's will become a 2; the second group of three 2's will become another 2; and the third group of three 2's will become yet another 2 (that's three 2's in all); also, the group of three 3's will become a 3. So the cube root of 13824 will be 2 x 2 x 2 x 3 = 24.
You can use the rules of exponents to see this also
(29 x 33 )1/3 =2(9/3) (3)(3/3) =23 3=24
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