Rationalise the denominator of each the following
`(i)(2)/(sqrt(3))" "(ii)(1)/(3sqrt(5))" "(iii)(1)/(sqrt(8))" "(iv)(sqrt(2)+1)/(sqrt(3))`

Explanation

Let's rationalize the denominators of the given expressions step by step.

### (i) Rationalize \( \frac{2}{\sqrt{3}} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{3} \):
\[ \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

**Final Answer:** \( \frac{2\sqrt{3}}{3} \)

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### (ii) Rationalize \( \frac{1}{3\sqrt{5}} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{5} \):
\[ \frac{1}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{3 \cdot 5} = \frac{\sqrt{5}}{15} \]

**Final Answer:** \( \frac{\sqrt{5}}{15} \)

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### (iii) Rationalize \( \frac{1}{\sqrt{8}} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{8} \):
\[ \frac{1}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{8}}{8} \]

**Final Answer:** \( \frac{\sqrt{8}}{8} \)

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### (iv) Rationalize \( \frac{\sqrt{2} + 1}{\sqrt{3}} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{3} \):
\[ \frac{\sqrt{2} + 1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}(\sqrt{2} + 1)}{3} \]

**Final Answer:** \( \frac{\sqrt{3}(\sqrt{2} + 1)}{3} \)

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