If x = √6 + √5 then find the value of x3– (1/x)3 =?
Solution
Correct Answer: Option C
x = √6 + √5
And \(\frac{1}{x} = \frac{1}{{\sqrt 6 {\rm{\;}} + {\rm{\;}}\sqrt 5 }}\)
\(= \;\frac{{\sqrt 6 \; - \;\sqrt 5 }}{{\left( {\sqrt 6 + \;\sqrt 5 } \right)\left( {\sqrt 6 \; - \;\sqrt 5 } \right)}}\) (Multiplying by \(\frac{{\sqrt 6 - \sqrt 5 }}{{\sqrt 6 - \sqrt 5 }}\) )
\(= \frac{{\sqrt 6 \; - \;\sqrt 5 }}{{6\; - \;5}}\)
= √6 - √5
∴ x – (1/x) = √6 + √5 - √6 + √5 = 2√5
∴ x3 – (1/x)3 = [x – (1/x)]3 + 3[x – (1/x)]
= (2√5)3 + 3 × 2√5
= 40√5 + 6√5
= 46√5
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