Question

$x = 3 + 2 \sqrt{2}$
Find the value of x³-1/x³​

Answer 1

Answer:

Given x= 3+2√2

we have to find the value of

${x}^{3} - \frac{1}{ {x}^{3} }$

first

1/x = 1/3+2√2

rationalizing the denominator

$= \frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2}) (3 - 2 \sqrt{2}) } \\ = \frac{3 - 2 \sqrt{2} }{9 - 8} = 3 - 2 \sqrt{2}$

now

x-1/x

=3+2√2-(3-2√2 )

=4√2

we know the algebraic identity which tells us

${a}^{3} - {b}^{3} = {(a - b)}^{3} + 3ab(a - b)$

now ,

${x}^{3} - \frac{1}{ {x}^{3} } \\ = {(4 \sqrt{2} )}^{3} + 3(4 \sqrt{2} ) \\ = 128 \sqrt{2} + 12 \sqrt{2} = 140 \sqrt{2}$

hope it helps you

Answer 2

Answer:

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