To solve the cube root of $-\frac{343}{1000}$, we will break it down step by step.

Step 1: Express the cube root operation

We need to find: $\sqrt[3]{-\frac{343}{1000}}$

Step 2: Separate the cube root into the numerator and the denominator

We can rewrite the expression as: [ \sqrt[3]{-\frac{343}{1000}} = \frac{\sqrt[3]{-343}}{\sqrt[3]{1000}} ]

Step 3: Calculate the cube root of the numerator

The numerator is $-343$. The cube root of $-343$ can be determined by recognizing that $343$ is $7^3$, so: [ \sqrt[3]{-343} = -\sqrt[3]{343} = -7 ] (negative because the cube root of a negative number is also negative).

Step 4: Calculate the cube root of the denominator

The denominator is $1000$, which is $10^3$, so: $\sqrt[3]{1000} = 10$

Step 5: Combine the results

Now, put the results of the numerator and the denominator together: [ \frac{\sqrt[3]{-343}}{\sqrt[3]{1000}} = \frac{-7}{10} = -\frac{7}{10} ]

Final Answer:

The cube root of $-\frac{343}{1000}$ is: $-\frac{7}{10} \text{ or } -0.7$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you solve the cube root of $-\frac{512}{729}$?
2. What is the cube root of $\frac{216}{125}$?
3. How do you simplify $\sqrt[3]{-27}$?
4. How do you solve $\sqrt[3]{-\frac{1}{8}}$?
5. How do you solve $\sqrt[3]{\frac{1}{64}}$?

Tip: Cube roots can be negative when the original number is negative, unlike square roots, which result in imaginary numbers for negative inputs.