Find the value of a and b in each of the following
`(i)(3+sqrt(2))/(3-sqrt(2))=a+bsqrt(2)" "(ii)(sqrt(2)+1)/(sqrt(2)-1)=a-bsqrt(2)" "(iii)(5+4sqrt(3))/(5-4sqrt(3))=a-bsqrt(3)`

Explanation

To find the values of \( a \) and \( b \) in the given expressions, we will simplify each expression step by step.

### (i) \(\frac{3 + \sqrt{2}}{3 - \sqrt{2}} = a + b\sqrt{2}\)

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator.

\[ \frac{(3 + \sqrt{2})(3 + \sqrt{2})}{(3 - \sqrt{2})(3 + \sqrt{2})} \]

**Step 2:** Simplify the numerator.

\[ (3 + \sqrt{2})^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{2} + (\sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2} \]

**Step 3:** Simplify the denominator.

\[ (3 - \sqrt{2})(3 + \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]

**Step 4:** Combine the results.

\[ \frac{11 + 6\sqrt{2}}{7} = \frac{11}{7} + \frac{6}{7}\sqrt{2} \]

**Step 5:** Compare with \( a + b\sqrt{2} \).

From the equation, we can see that:

\[ a = \frac{11}{7}, \quad b = \frac{6}{7} \]

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### (ii) \(\frac{\sqrt{2} + 1}{\sqrt{2} - 1} = a - b\sqrt{2}\)

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator.

\[ \frac{(\sqrt{2} + 1)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} \]

**Step 2:** Simplify the numerator.

\[ (\sqrt{2} + 1)^2 = (\sqrt{2})^2 + 2 \cdot \sqrt{2} \cdot 1 + 1^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]

**Step 3:** Simplify the denominator.

\[ (\sqrt{2} - 1)(\sqrt{2} + 1) = (\sqrt{2})^2 - 1^2 = 2 - 1 = 1 \]

**Step 4:** Combine the results.

\[ \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2} \]

**Step 5:** Compare with \( a - b\sqrt{2} \).

From the equation, we can see that:

\[ a = 3, \quad b = -2 \]

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### (iii) \(\frac{5 + 4\sqrt{3}}{5 - 4\sqrt{3}} = a - b\sqrt{3}\)

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator.

\[ \frac{(5 + 4\sqrt{3})(5 + 4\sqrt{3})}{(5 - 4\sqrt{3})(5 + 4\sqrt{3})} \]

**Step 2:** Simplify the numerator.

\[ (5 + 4\sqrt{3})^2 = 5^2 + 2 \cdot 5 \cdot 4\sqrt{3} + (4\sqrt{3})^2 = 25 + 40\sqrt{3} + 48 = 73 + 40\sqrt{3} \]

**Step 3:** Simplify the denominator.

\[ (5 - 4\sqrt{3})(5 + 4\sqrt{3}) = 5^2 - (4\sqrt{3})^2 = 25 - 48 = -23 \]

**Step 4:** Combine the results.

\[ \frac{73 + 40\sqrt{3}}{-23} = -\frac{73}{23} - \frac{40}{23}\sqrt{3} \]

**Step 5:** Compare with \( a - b\sqrt{3} \).

From the equation, we can see that:

\[ a = -\frac{73}{23}, \quad b = \frac{40}{23} \]

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### Summary of Results:
1. \( a = \frac{11}{7}, b = \frac{6}{7} \)
2. \( a = 3, b = -2 \)
3. \( a = -\frac{73}{23}, b = \frac{40}{23} \)

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