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Question
If x = `(7 + 4sqrt(3))`, find the value of `x^3 + (1)/x^3`.
Sum
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Solution
`x^3 + (1)/x^3`
`(x^3 + (1)/x^3) = (x + (1)/x)^3 - 3(x + (1)/x)` ...(1)
We will first find out `x + (1)/x`
`x + (1)/x = (7 + 4sqrt(3)) + (1)/((7 + 4sqrt(3))`
= `((7 + 4sqrt(3))^2 + 1)/((7 + 4sqrt(3))`
= `(49 + 48 + 56sqrt(3) + 1)/((7 + 4sqrt(3))`
= `(98 + 56sqrt(3))/((7 + 4sqrt(3))`
= `(14(7 + 4sqrt(3)))/((7 + 4sqrt(3))`
= 14
Substituting in (1)
`(x^3 + (1)/x^3) = (x + (1)/x)^3 - 3(x + (1)/x)`
= (14)3 – 3 × 14
= 2744 – 42
= 2702
∴ `(x^3 + (1)/x^3) = 2702`
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Simplifying an Expression by Rationalization of the Denominator
Is there an error in this question or solution?
Chapter 3: Expansions - Exercise 3B [Page 73]
