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Question
If x = `(7 + 4sqrt(3))`, find the value of `x^3 + (1)/x^3`.
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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`
RELATED QUESTIONS
Rationalize the denominator.
`4/(7+ 4 sqrt3)`
Rationalise the denominators of : `[ 2 - √3 ]/[ 2 + √3 ]`
Rationalise the denominators of : `[ sqrt3 - sqrt2 ]/[ sqrt3 + sqrt2 ]`
Simplify by rationalising the denominator in the following.
`(1)/(5 + sqrt(2))`
Simplify by rationalising the denominator in the following.
`(3 - sqrt(3))/(2 + sqrt(2)`
Simplify by rationalising the denominator in the following.
`(5 + sqrt(6))/(5 - sqrt(6)`
Simplify the following
`(sqrt(7) - sqrt(3))/(sqrt(7) + sqrt(3)) - (sqrt(7) + sqrt(3))/(sqrt(7) - sqrt(3)`
If x = `(4 - sqrt(15))`, find the values of
`x^3 + (1)/x^3`
If x = `(1)/((3 - 2sqrt(2))` and y = `(1)/((3 + 2sqrt(2))`, find the values of
x2 + y2
Show that:
`(4 - sqrt5)/(4 + sqrt5) + 2/(5 + sqrt3) + (4 + sqrt5)/(4 - sqrt5) + 2/(5 - sqrt3) = 52/11`
