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Even numbered exponents of any real number generate a positive result. Odd numbered exponents of a real number generate a result with the same sign as the real number. This is because + times + and - times - both produce a positive result ((+1)²=(-1)²) but minus times itself an odd number of times is minus ((+1)²ⁿ⁺¹=(+1)²ⁿ×(+1)=+1, (-1)²ⁿ⁺¹=(-1)²ⁿ×(-1)=-1, where n is an integer).

64=8×8=(-8)×(-8), but 64=4×4×4 and -64=(-4)×(-4)×(-4). Therefore, √64 is 8 or -8, while ∛64=4 and ∛(-64)=-∛64=-4.

In the world of complex numbers, all numbers have 2 square roots and 3 cube roots. In this world, 64 and -64 each have 3 cube roots, but two of them are complex in each case.

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