Question

$\frac{1}{ \sqrt{6} - \sqrt{5}} - \frac{1}{ \sqrt{5} - \sqrt{4}} + \frac{1}{ \sqrt{4} - \sqrt{3} }$
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Answer 1

Answer:

4 + √6 - √3

Solution:

We need to find the value of ;

1/(√6 - √5) - 1/(√5 - √4) + 1/(√4 - √3)

Rationalising the first term, we get ;

1/(√6 - √5) = (√6 + √5)/(√6 - √5)(√6 + √5)

= (√6 + √5)/[(√6)² - (√5)²]

= (√6 + √5)/(6 - 5)

= (√6 + √5)/1

= √6 + √5

Rationalising the second term, we get ;

1/(√5 - √4) = (√5 + √4)/(√5 - √4)(√5 + √4)

= (√5 + √4)/[(√5)² - (√4)²]

= (√5 + √4)/(5 - 4)

= (√5 + √4)/1

= √5 + √4

Rationalising the third term, we get ;

1/(√4 - √3) = (√4 + √3)/(√4 - √3)(√4 + √3)

= (√4 + √3)/[(√4)² - (√3)²]

= (√4 + √3)/(4 - 3)

= (√4 + √3)/1

= √4 + √3

Thus,

1/(√6 - √5) - 1/(√5 - √4) + 1/(√4 - √3)

= (√6 + √5) - (√5 - √4) + (√4 - √3)

= √6 + √5 - √5 + √4 + √4 - √3

= √6 + 2√4 - √3

= √6 + 2×2 - √3

= √6 + 4 - √3

= 4 + √6 - √3

Hence,

The required value of

1/(√6 - √5) - 1/(√5 - √4) + 1/(√4 - √3) is

4 + √6 - √3 .