Solve the following :-
(i)`sqrt(sqrt(2)/(3))xxsqrt((9)/(2))`
(ii)`sqrt((5)/(6))xxsqrt((3)/(5))`
(iii)`sqrt((3)/(5))divsqrt((27)/(125))`
(iv)`sqrt((4)/(9))divsqrt((64)/(81))`

Answer

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

### Part (i):
Solve \( \sqrt{\frac{\sqrt{2}}{3}} \times \sqrt{\frac{9}{2}} \)

**Step 1:** Rewrite the expression using properties of square roots.
\[ \sqrt{\frac{\sqrt{2}}{3}} = \frac{\sqrt{2}}{\sqrt{3}} \quad \text{and} \quad \sqrt{\frac{9}{2}} = \frac{\sqrt{9}}{\sqrt{2}} = \frac{3}{\sqrt{2}} \]

**Step 2:** Multiply the two fractions.
\[ \frac{\sqrt{2}}{\sqrt{3}} \times \frac{3}{\sqrt{2}} = \frac{3 \cdot \sqrt{2}}{\sqrt{3} \cdot \sqrt{2}} = \frac{3}{\sqrt{3}} \]

**Step 3:** Simplify the fraction.
\[ \frac{3}{\sqrt{3}} = \sqrt{3} \quad \text{(since } \frac{3}{\sqrt{3}} = \sqrt{3} \text{)} \]

**Final Answer for Part (i):** \( \sqrt{3} \)

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### Part (ii):
Solve \( \sqrt{\frac{5}{6}} \times \sqrt{\frac{3}{5}} \)

**Step 1:** Rewrite the expression using properties of square roots.
\[ \sqrt{\frac{5}{6}} = \frac{\sqrt{5}}{\sqrt{6}} \quad \text{and} \quad \sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}} \]

**Step 2:** Multiply the two fractions.
\[ \frac{\sqrt{5}}{\sqrt{6}} \times \frac{\sqrt{3}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{6}} \]

**Step 3:** Simplify the fraction.
\[ \frac{\sqrt{3}}{\sqrt{6}} = \frac{\sqrt{3}}{\sqrt{3 \cdot 2}} = \frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{2}} = \frac{1}{\sqrt{2}} \]

**Final Answer for Part (ii):** \( \frac{1}{\sqrt{2}} \)

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### Part (iii):
Solve \( \sqrt{\frac{3}{5}} \div \sqrt{\frac{27}{125}} \)

**Step 1:** Rewrite the division as multiplication by the reciprocal.
\[ \sqrt{\frac{3}{5}} \div \sqrt{\frac{27}{125}} = \sqrt{\frac{3}{5}} \times \sqrt{\frac{125}{27}} \]

**Step 2:** Multiply the two fractions.
\[ \frac{\sqrt{3}}{\sqrt{5}} \times \frac{5}{3} = \frac{\sqrt{3} \cdot 5}{\sqrt{5} \cdot 3} \]

**Step 3:** Simplify the expression.
\[ = \frac{\sqrt{3}}{3} \quad \text{(since } \frac{5}{\sqrt{5}} = \sqrt{5} \text{)} \]

**Final Answer for Part (iii):** \( \frac{\sqrt{3}}{3} \)

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### Part (iv):
Solve \( \sqrt{\frac{4}{9}} \div \sqrt{\frac{64}{81}} \)

**Step 1:** Rewrite the division as multiplication by the reciprocal.
\[ \sqrt{\frac{4}{9}} \div \sqrt{\frac{64}{81}} = \sqrt{\frac{4}{9}} \times \sqrt{\frac{81}{64}} \]

**Step 2:** Simplify the square roots.
\[ \sqrt{\frac{4}{9}} = \frac{2}{3} \quad \text{and} \quad \sqrt{\frac{81}{64}} = \frac{9}{8} \]

**Step 3:** Multiply the two fractions.
\[ \frac{2}{3} \times \frac{9}{8} = \frac{2 \cdot 9}{3 \cdot 8} = \frac{18}{24} \]

**Step 4:** Simplify the fraction.
\[ \frac{18}{24} = \frac{3}{4} \]

**Final Answer for Part (iv):** \( \frac{3}{4} \)

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### Summary of Answers:
1. \( \sqrt{3} \)
2. \( \frac{1}{\sqrt{2}} \)
3. \( \frac{\sqrt{3}}{3} \)
4. \( \frac{3}{4} \)

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