Calculate (2/3)³: Finding the Cube of Two-Thirds

To solve this problem, we'll follow these steps:

• Step 1: Identify the given fraction, which is $\frac{2}{3}$.
• Step 2: Apply the formula for exponents applied to fractions, $\displaystyle(\frac{a}{b})^n = \frac{a^n}{b^n}$.
• Step 3: Calculate the cube of the numerator and the cube of the denominator separately.
• Step 4: Write the results as a single fraction.

Now, let's work through each step:

Step 1: The problem provides the fraction $\frac{2}{3}$.

Step 2: Use the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$ for $a = 2$, $b = 3$, and $n = 3$.

Step 3: We need to calculate $2^3$ and $3^3$:
- Calculate $2^3 = 2 \times 2 \times 2 = 8$.
- Calculate $3^3 = 3 \times 3 \times 3 = 27$.

Step 4: Writing these as a fraction gives $\displaystyle \frac{2^3}{3^3} = \frac{8}{27}$.

Therefore, the solution to the problem is $\frac{8}{27}$.