Question

if x=2+√3, find the value
of x3

Answer 1

Heya !!

Here's your answer.. ⬇⬇
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⚫ Given :- x = 2 + √3

⚫ To Find :- x³ + 1/x³ = ?

⚫ Solution :- x = 2 + √3

$\frac{1}{x} = \frac{1}{2 + \sqrt{3} } \\ \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \frac{1}{x} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{x} = 2 - \sqrt{3} \\ \\ x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} \\ \\ x + \frac{1}{x} = 4 \\ \\ take \: \: cube \: \: on \: \: both \: \: side \\ \\ {(x + \frac{1}{x} )}^{3} = {(4)}^{3} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3x \times \frac{1}{x} (x + \frac{1}{x} ) = 64 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3(4) = 64 \: \: \: \: \: \:..... (x + \frac{1}{x} = 4) \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 64 - 12 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 52$
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Hope it helps..

Thanks :))

Answer 2

hope it helps to you.