Question

if a-1/a=√5 ,find the value ofa+1/a and a^3+1/a^3​

Answer 1

Question  -

if a-1/a=√5 ,find the value of a+1/a and a^3+1/a^3​ .

Solution -

$a \ - \dfrac{1}{a} =3 \\ \\ Squaring \ Both \ Sides \\ \\a^{2} + \dfrac{1}{ a^2} = 9 + 2 = 11 \\ \ \\ ( a \ + \ \dfrac{1}{a} ) ^2 = 11 + 2 = 13 \\\\\\\therefore a \ + \ \dfrac{1}{a}= \sqrt{13} \ ..... [A_{1}] \\\\\\ Cubing \ Both \ Sides \\ \\ \\ a^3 \ + \ \ ( \dfrac{1}{a} )^3 + 3\sqrt{13} = 3\sqrt{13} \\ \\ \\ \leadsto  a^3 \ + \ \ ( \dfrac{1}{a} )^3 = 0 \ ........ [ A_{2}]$

Answer 2

Step-by-step explanation:

a-1/a=√5

(a-1/a)²=5

(a+1/a)²-4=5

(a+1/a)²=9

a+1/a=±3

when, a+1/a=3

a³+1/a³

=(a+1/a)³-3(a+1/a)

=(3)³-3(3)

=27-9

=18

when,a+1/a=-3

a³+1/a³

=(a+1/a)³-3(a+1/a)

=(-3)³-3(-3)

=-27+9

=-18