Question

If (a-1/a)=√5 find the values of (i) (a+1/a) (ii) (a^3+1/a^3)​

Answer 1

Answer:

(i) 3

(ii) 18

Step-by-step explanation:

Given that ;

$\sf \left(a - \dfrac{1}{a} \right) = \sqrt{5}$

On squaring both sides ;

$\sf \left(a - \frac{1}{a} \right) {}^{2} = ( \sqrt{5}) {}^{2} \\ \\ \implies \sf {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 5 \\ \\ \implies \sf {a}^{2} + \frac{1}{ {a}^{2} } = 5 + 2 \\ \\ \implies \sf {a}^{2} + \frac{1}{ {a}^{2} } = 7$

On adding 2 both sides ;

$\implies \sf {a}^{2} + \frac{1}{a {}^{2} } + 2 = 7 +2 \\ \\ \implies \sf {a}^{2} + \frac{1}{ {a}^{2}} + 2 \: . \: a\: .\: \frac{1}{a} = 9 \\ \\ \implies \sf \left(a + \frac{1}{a} \right) {}^{2} = (3) {}^{2} \\ \\ \boxed{ \red{ \therefore \: \bf a + \frac{1}{a} = 3}}$

On cubing both sides ;

$\implies \sf \left( a + \frac{1}{a} \right)^{3} = (3) {}^{3} \\ \\ \implies \sf {a}^{3} + \frac{1}{ {a}^{3} } + 3 \: .\: a \: . \: \frac{1}{a} (a + \frac{1}{a} ) = 27 \\ \\ \implies \sf {a}^{3} + \frac{1}{ {a}^{3} } + 3(3) = 27 \\ \\ \implies \sf {a}^{3} + \frac{1}{ {a}^{3} } + 9 = 27 \\ \\ \implies \sf {a}^{3} + \frac{1}{ {a}^{3} } = 27 - 9 \\ \\ \boxed{ \red{ \therefore \: \bf {a}^{3} + \frac{1}{ {a}^{3} } = 18}}$

Answer 2

Answer:

pls see the attachment