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Question

In the following determine rational numbers a and b: $$\dfrac{5+3\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$$

A
$$ a = 1, b = 1$$
B
$$ a = -1, b = -1$$
C
$$ a = -1, b = 1$$
D
None of the above
Solution
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Correct option is B. $$ a = -1, b = -1$$
Given,

$$\dfrac{5+3\sqrt{3}}{7+4\sqrt{3}}$$

$$=\dfrac{\left(5+3\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}$$

$$=\dfrac{\sqrt{3}-1}{1}$$

$$=-1+\sqrt 3$$

$$=a+b\sqrt 3$$

upon comparision, we get,

$$a=-1,b=1$$

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