Simplify :
`((8 x^(3)) /(125y^(3)))^((2)/(3))`

Answer

To simplify the expression \(\left(\frac{8x^3}{125y^3}\right)^{\frac{2}{3}}\), we will follow these steps:

### Step 1: Rewrite the base
First, we can express \(8\) and \(125\) in terms of their prime factors:
- \(8 = 2^3\)
- \(125 = 5^3\)

Thus, we can rewrite the expression as:
\[ \left(\frac{2^3 x^3}{5^3 y^3}\right)^{\frac{2}{3}} \]

### Step 2: Apply the exponent to the numerator and denominator
Using the property of exponents \((\frac{a}{b})^n = \frac{a^n}{b^n}\), we can apply the exponent \(\frac{2}{3}\) to both the numerator and the denominator:
\[ \frac{(2^3 x^3)^{\frac{2}{3}}}{(5^3 y^3)^{\frac{2}{3}}} \]

### Step 3: Simplify the numerator and denominator
Now we simplify both the numerator and the denominator:
- For the numerator:
\[ (2^3 x^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} x^{3 \cdot \frac{2}{3}} = 2^2 x^2 = 4x^2 \]
- For the denominator:
\[ (5^3 y^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}} y^{3 \cdot \frac{2}{3}} = 5^2 y^2 = 25y^2 \]

### Step 4: Combine the results
Now we can combine the simplified numerator and denominator:
\[ \frac{4x^2}{25y^2} \]

### Final Answer
Thus, the simplified form of the expression \(\left(\frac{8x^3}{125y^3}\right)^{\frac{2}{3}}\) is:
\[ \frac{4x^2}{25y^2} \]

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