solve cube root of -343/1000 step by step explanation

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.

Learn how to solve the cube root of -343/1000 with a detailed step-by-step explanation. This problem covers cube roots, including the cube root operation, calculation of cube roots of specific numbers (-343 and 1000), and understanding cube roots of negative numbers. Suitable for students in Grades 7-9.

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