Question

solve the following question.

Answer 1

Question:-

Find the value of a and b if

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

Solution:-

$\frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} }$

Take a lcm, we get

$\frac{(3 - \sqrt{5} )(7 + 3 \sqrt{5}) - (3 + \sqrt{5} )(7 - 3 \sqrt{5} )}{(3 + \sqrt{5} )(3 - \sqrt{5} )}$

Use this identity

=> (a-b)(c+d)=ac+ad-bc-bd

=>(a-b)(a+b)=(a²-b²)

By using this Identity,we get

$\frac{(21 + 9 \sqrt{5} - 7 \sqrt{5} - 15) - (21 - 9 \sqrt{5} + 7 \sqrt{5} - 15 )}{9 - 5}$

$\frac{21 + 9 \sqrt{5} - 7 \sqrt{5} - 15 - 21 + 9 \sqrt{5} - 7 \sqrt{5} + 15}{4}$

$\frac{2 \sqrt{5} + 2 \sqrt{5} }{4}$

$\frac{4 \sqrt{5} }{4}$

$\sqrt{5}$

Now we can write as

$0 + 1 \sqrt{5}$

Answer:- a=0 and b=1

Answer 2

Answer:

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Step-by-step explanation:

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