Question

simplify the following
$\frac{ \sqrt{5 - 2 } }{ \sqrt{5 + 2} } - \frac{ \sqrt{5 + 2} }{ \sqrt{5 - 2} }$
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Answer 1

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Answer 2

Step-by-step explanation:

Correction:-

Given that :

We have,

[(√5-2)/(√5+2)] - [(√5+2)/(√5-2)]

First term:

[(√5-2)/(√5+2)]

=> [(√5-2)/(√5+2)] × [(√5-2)/(√5-2)]

=> [(√5-2)(√5-2)/(√5+2)(√5-2)]

=> (√5-2)^2/[(√5+2)/(√5-2)]

=> (√5-2)^2/[(√5)^2-(2)^2]

=> (√5-2)²/(5-4)

=> (√5-2)²/1

=> (√5-2)²

It is in the form of (a-b)²

Where, a = √5 and b = 2

(a-b)^2 = a^2-2ab+b^2

=> (√5)^2 -2(√5)(2)+(2)^2

=> 5-4√5+4

=> 9-4√5

[(√5-2)/(√5+2)] = 9-4√5 ------------(1)

Second term:

[(√5+2)/(√5-2)]

=> [(√5+2)/(√5-2)] × [(√5+2)/(√5+2)]

=> [(√5+2)(√5+2)/(√5-2)(√5+2)]

=> (√5+2)^2/[(√5+2)/(√5-2)]

=> (√5+2)^2/[(√5)^2-(2)^2]

=> (√5+2)^2/(5-4)

=> (√5+2)^2/1

=> (√5+2)^2

It is in the form of (a+b)^2

Where, a = √5 and b = 2

(a+b)² = a^2+2ab+b^2

=> (√5)^2+2(√5)(2)+(2)^2

=> 5+4√5+4

=> 9+4√5

[(√5+2)/(√5-2)] = 9+4√5 ----------(2)

Now,

[(√5-2)/(√5+2)] - [(√5+2)/(√5-2)]

From (1)&(2)

=> (9-4√5)-(9+4√5)

=> 9-4√5-9-4√5

=> (9-9)-(4√5+4√5)

=> 0-(8√5)

=> -8√5

Hence, the denominator is rationalised.