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The problem is asking to explain why:

\[
\sqrt[3]{\frac{8}{27}} = \frac{2}{3}
\]

Let's break it down step by step:

### Step 1: Simplifying the Cube Root
We are tasked with taking the cube root of a fraction. The cube root of a fraction is the same as taking the cube root of the numerator and the denominator separately:

\[
\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}}
\]

### Step 2: Cube Roots of the Numerator and Denominator
Now, let's calculate the cube roots of the numerator (8) and the denominator (27).

- The cube root of 8 is:
  \[
  \sqrt[3]{8} = 2 \quad \text{(since } 2^3 = 8\text{)}
  \]

- The cube root of 27 is:
  \[
  \sqrt[3]{27} = 3 \quad \text{(since } 3^3 = 27\text{)}
  \]

### Step 3: Form the Fraction
Now, substitute the cube roots back into the fraction:

\[
\frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}
\]

Thus, the cube root of \( \frac{8}{27} \) simplifies to \( \frac{2}{3} \), which explains why:

\[
\sqrt[3]{\frac{8}{27}} = \frac{2}{3}
\]

Would you like more details on any specific step, or do you have any questions?

Here are 5 related questions to explore further:
1. How would you simplify \( \sqrt[3]{\frac{64}{125}} \)?
2. What is the cube root of a negative number, for example \( \sqrt[3]{-8} \)?
3. Can cube roots always be simplified to exact fractions like in this example?
4. How does the process differ when finding a square root versus a cube root?
5. What is the general rule for taking the \( n \)-th root of a fraction?

**Tip:** Remember that when simplifying cube roots, you can always apply the cube root to the numerator and denominator separately.

Explain why the cube root of 8/27 is 2/3.

Learn how to simplify the cube root of 8/27 step by step. This problem involves calculating cube roots of both the numerator and denominator, applying cube root properties, and understanding how fractions work in simplification. Suitable for Grades 7-9.

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