How many real third roots does -343 have? Explain your reasoning.

One root

Explanation:

x^3 = -343

x^3 + 343 = 0

x^3 + 7^3 = 0

(x+7)(x^2 - 7x + 7^2) = 0 sum of cubes factoring rule

(x+7)(x^2 - 7x + 49) = 0

Set each piece equal to zero and solve for x.

You should find that the first piece x+7 = 0 leads to x = -7 as the only real root.

The other piece x^2-7x+49 = 0 leads to two non-real roots, ie complex roots.