Question

if a=7-4√3find the value of a3+1/a3

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Answer 1

Step-by-step explanation:

⚫ Given :- a = 2 + √3

⚫ To Find :- a³ + 1/a³ = ?

⚫ Solution :- a = 2 + √3

\frac{1}{a} = \frac{1}{2 + \sqrt{3} } \\ \\ \frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{a} = 2 - \sqrt{3} \\ \\ a + \frac{1}{a} = 2 + \sqrt{3} + 2 - \sqrt{3} \\ \\ a + \frac{1}{a} = 4 \\ \\ take \: \: cube \: \: both \: \: side \: \: \\ \\ {(a + \frac{1}{a} )}^{3} = {4}^{3} \\ \\ {a}^{3} + \frac{1}{ {a}^{3} } + 3a \times \frac{1}{a} (a + \frac{1}{a} ) = 64 \\ \\ {a}^{3} + \frac{1}{ {a}^{3} } + 3(4) = 64 \\ \\ {a}^{3} + \frac{1}{ {a}^{3} } = 64 - 12 \\ \\ {a}^{3} + \frac{1}{ {a}^{3} } = 52