Question

If
$x = \sqrt{3} - \sqrt{2}$
find the value of
${x}^{3} - (1 \div {x}^{3} )$
Use the formula

${x}^{3} - 1 \: by \: {x}^{3} - 3x \times 1 \div x( x - 1 \div x)$
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Answer 1

Given :   x = √3 - √2

To Find :  Value of  x³  - 1/x³

Solution:

x = √3 - √2

1/x = 1/(√3 - √2)

on rationalizing

=> 1/x = (√3 + √2)/(3 - 2)

=> 1/x = √3 + √2

(a - b)³  = a³ - b³  - 3ab(a - b)

a = x  b = 1/x

=> (x - 1/x)³ = x³ - 1/x³  - 3x(1/x)(x - 1/x)

x = √3 - √2

1/x = √3 + √2

=> x - 1/x  = (√3 - √2 ) - ( √3 + √2)  = -2√2

(x - 1/x)³ = x³ - 1/x³  - 3x(1/x)(x - 1/x)

=> (-2√2)³  = x³ - 1/x³  - 3(-2√2)

=> -16√2  = x³ - 1/x³  +6√2

=>  x³ - 1/x³  = -22√2

 x³ - 1/x³  = -22√2

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