Question

find a and b = 5+2√3 by 7+4√3=a-b√3

Answer 1

the answer this question is see in photo

Answer 2

Hi friend, Harish here.

Here is your answer:

Given that,

$\frac{5+2 \sqrt{3} }{7+4 \sqrt{3} } = a-b \sqrt{3}$

To find,

The value of a & b.

Solution,

First we must rationalize the denominator by multiplying and dividing the number by it's conjugate.

Conjugate is 7 - 4√ 3 .

Then,

$\frac{5+2 \sqrt{3} }{7+4 \sqrt{3} } \times \frac{7-4 \sqrt{3}}{7-4 \sqrt{3}} = a-b \sqrt{3}$

Here to multiply the denominator we must use the identity:

(x+y) (x-y) = x² - y².

Then,

(7 + 4√3) (7 - 4√3) = (7)² - (4√3)² = 49 - 48 = 1.

So,

$\frac{5+2 \sqrt{3} }{7+4 \sqrt{3} } \times \frac{7-4 \sqrt{3}}{7-4 \sqrt{3}} = \frac{(35 +14 \sqrt{3} -20 \sqrt{3} - 24 ) }{(7+4 \sqrt{3}(7-4 \sqrt{3}) } = \frac{11-6 \sqrt{3} }{1}$

⇒ $11-6 \sqrt{3} = a-b \sqrt{3}$

Now by comparing we get,

a = 11 & b = 6.
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Hope my  answer is helpful to you.