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.