The expression given in the image is:

$\sqrt[3]{-343y^6}$

To find the cube root of this expression, we can break it down into two parts:

1. The cube root of $-343$
2. The cube root of $y^6$

Step 1: Cube root of $-343$

$\sqrt[3]{-343} = -7$ because $(-7)^3 = -343$.

Step 2: Cube root of $y^6$

The cube root of $y^6$ is: $\sqrt[3]{y^6} = y^2$ because $y^6 = (y^2)^3$.

Final Answer:

Combining these results, we get: $\sqrt[3]{-343y^6} = -7y^2$

Would you like further details or explanations?

Here are 5 related questions to explore further:

1. How do you find the cube root of negative numbers?
2. What is the general rule for taking the cube root of powers of variables?
3. What is the difference between cube roots and square roots in terms of handling negative numbers?
4. How can you simplify expressions involving cube roots and higher powers of variables?
5. What happens if you need to find the cube root of more complex algebraic expressions?

Tip: When simplifying cube roots, remember to break the problem into smaller parts: first handle constants, then deal with the variables.