Question

if a = 2-root3 find [a-1/a]³





please answer​

Answer 1

Solution:

Given,

$\tt \longrightarrow a = 2 - \sqrt{3}$

Therefore,

$\tt \longrightarrow \dfrac{1}{a} = \dfrac{1}{2 - \sqrt{3}}$

$\tt \longrightarrow \dfrac{1}{a} = \dfrac{1}{2 - \sqrt{3}} \times \dfrac{2 + \sqrt{3} }{2 + \sqrt{3} }$

$\tt \longrightarrow \dfrac{1}{a} = \dfrac{2 + \sqrt{3} }{ {2}^{2} - ( \sqrt{3})^{2} }$

$\tt \longrightarrow \dfrac{1}{a} = \dfrac{2 + \sqrt{3} }{4 - 3}$

$\tt \longrightarrow \dfrac{1}{a} = 2 + \sqrt{3}$

Therefore,

$\tt \longrightarrow a - \dfrac{1}{a} = 2 - \sqrt{3} - (2 + \sqrt{3})$

$\tt \longrightarrow a - \dfrac{1}{a} = 2 - \sqrt{3} - 2 - \sqrt{3}$

$\tt \longrightarrow a - \dfrac{1}{a} = - 2\sqrt{3}$

Cubing both sides, we get,

$\tt \longrightarrow \bigg(a - \dfrac{1}{a} \bigg)^{3} =( - 2\sqrt{3})^{3}$

$\tt \longrightarrow \bigg(a - \dfrac{1}{a} \bigg)^{3} = - 24 \sqrt{3}$

Which is our required answer.

Answer:

• (a - 1/a)³ = -24√3.

Learn More:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + 3ab(a + b) + b³
• (a - b)³ = a³ - 3ab(a - b) - b³
• a³ + b³ = (a + b)(a² - ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)