Evaluate:
`(i) ((2)/(3))^(3) xx ((2)/(3)) ^(2)" "(ii)((4)/(7))^(7)xx ((4)/(7))^(-3)" "(iii) ((3)/(2))^(-3) " "(iv) ((8)/(5))^(-3) xx((8)/(5))^(2)`

Answer

Let's evaluate each part of the question step by step.

### (i) Evaluate \(\left(\frac{2}{3}\right)^{3} \times \left(\frac{2}{3}\right)^{2}\)

**Step 1:** Identify the bases and add the exponents.
- The base is \(\frac{2}{3}\).
- The exponents are \(3\) and \(2\).
- According to the property of exponents, when multiplying like bases, we add the exponents:
\[ \left(\frac{2}{3}\right)^{3 + 2} = \left(\frac{2}{3}\right)^{5} \]

**Step 2:** Calculate \(\left(\frac{2}{3}\right)^{5}\).
- This means:
\[ \left(\frac{2}{3}\right)^{5} = \frac{2^{5}}{3^{5}} = \frac{32}{243} \]

**Final Answer for (i):** \(\frac{32}{243}\)

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### (ii) Evaluate \(\left(\frac{4}{7}\right)^{7} \times \left(\frac{4}{7}\right)^{-3}\)

**Step 1:** Identify the bases and add the exponents.
- The base is \(\frac{4}{7}\).
- The exponents are \(7\) and \(-3\).
- Adding the exponents:
\[ \left(\frac{4}{7}\right)^{7 + (-3)} = \left(\frac{4}{7}\right)^{4} \]

**Step 2:** Calculate \(\left(\frac{4}{7}\right)^{4}\).
- This means:
\[ \left(\frac{4}{7}\right)^{4} = \frac{4^{4}}{7^{4}} = \frac{256}{2401} \]

**Final Answer for (ii):** \(\frac{256}{2401}\)

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### (iii) Evaluate \(\left(\frac{3}{2}\right)^{-3}\)

**Step 1:** Apply the property of negative exponents.
- The base is \(\frac{3}{2}\) and the exponent is \(-3\).
- According to the property:
\[ \left(\frac{3}{2}\right)^{-3} = \frac{2^{3}}{3^{3}} \]

**Step 2:** Calculate \(\frac{2^{3}}{3^{3}}\).
- This means:
\[ \frac{2^{3}}{3^{3}} = \frac{8}{27} \]

**Final Answer for (iii):** \(\frac{8}{27}\)

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### (iv) Evaluate \(\left(\frac{8}{5}\right)^{-3} \times \left(\frac{8}{5}\right)^{2}\)

**Step 1:** Identify the bases and add the exponents.
- The base is \(\frac{8}{5}\).
- The exponents are \(-3\) and \(2\).
- Adding the exponents:
\[ \left(\frac{8}{5}\right)^{-3 + 2} = \left(\frac{8}{5}\right)^{-1} \]

**Step 2:** Apply the property of negative exponents.
- This means:
\[ \left(\frac{8}{5}\right)^{-1} = \frac{5}{8} \]

**Final Answer for (iv):** \(\frac{5}{8}\)

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### Summary of Answers:
1. \(\frac{32}{243}\)
2. \(\frac{256}{2401}\)
3. \(\frac{8}{27}\)
4. \(\frac{5}{8}\)

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