The problem asks us to simplify $(-343)^{\frac{1}{3}}$, which means we need to find the cube root of $-343$.

Solution:

1. The cube root of a number $x$ is the number $y$ such that $y^3 = x$.
2. For $-343$, the cube root is $-7$, because: $(-7)^3 = (-7) \cdot (-7) \cdot (-7) = -343$

Final Answer:

$(-343)^{\frac{1}{3}} = -7$

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Here are 5 related questions for practice:

1. Simplify $(-27)^{\frac{1}{3}}$.
2. Evaluate $8^{\frac{1}{3}}$.
3. Simplify $(64)^{\frac{2}{3}}$.
4. What is $(-125)^{\frac{1}{3}}$?
5. Solve $x^3 = -512$ for $x$.

Tip: For any number $a$, $a^{\frac{1}{3}}$ is the same as finding the cube root of $a$.