Question

if a=5-2√6,then find the value of a^3+1/a^3​

Answer 1

Answer:

$\large{\underline{\boxed{\sf 970}}}$

Step-by-step explanation:

Given that ;

a = 5 - 2√6

Now, find the value of 1/a.

$\implies \sf \frac{1}{a} = \frac{1}{5 - 2 \sqrt{6} } \\ \\ \implies \sf \frac{1}{a} = \frac{1}{5 - 2 \sqrt{6} } \times \frac{5 + 2 \sqrt{6} }{5 + 2 \sqrt{6} } \\ \\ \implies \sf \frac{1}{a} = \frac{5 + 2 \sqrt{6} }{(5)^{2} - (2 \sqrt{6}) {}^{2} } \\ \\ \implies \sf \frac{1}{a} = \frac{5 + 2 \sqrt{6} }{25 - 24} \\ \\ \implies \sf \frac{1}{a} = 5 + 2 \sqrt{6}$

Find the value of a + (1/a).

$\implies \sf a + \frac{1}{a} = 5 - 2 \sqrt{6} + 5 - 2 \sqrt{6} \\ \\ \implies \sf a + \frac{1}{a} = 10$

On cubing both sides ;

$\implies \sf (a + \frac{1}{a}) {}^{3} = (10) {}^{3} \\ \\ \implies \sf a ^{3} + \frac{1}{a^{3}} + 3 \:. \: a\: .\: \frac{1}{a}(a + \frac{1}{a}) = 1000 \\ \\ \implies \sf a ^{3} + \frac{1}{a^{3}}+ 3(10) = 1000 \\ \\ \implies \sf a ^{3} + \frac{1}{a^{3}} + 30 = 1000 \\ \\\implies \sf a ^{3} + \frac{1}{a^{3}} = 1000 - 30 \\ \\ \red{\boxed{ \bf \therefore \: a ^{3} + \frac{1}{a^{3}} = 970}}$