Question

if a not=0 and a-1/a=3 find

(i) a2+1/a2

(ii) a3-1/a3

Answer 1

Answer :

Given a $\ne$ 0 And

a - $\frac{1}{a}$ = 3 --------------- ( 1 )

i ) Find value of : a² + $\frac{1}{a^2}$


Squaring equation 1 , we get

( a - $\frac{1}{a}$ )² = 3²

$\Rightarrow$a² + $\frac{1}{a^2}$ - 2 ( a ) ( $\frac{1}{a}$ ) = 9 [ As we know ( a + b )² = a² + b² + 2ab ]

$\Rightarrow$a² + $\frac{1}{a^2}$ - 2 = 9

$\Rightarrow$a² + $\frac{1}{a^2}$ = 9 + 2

$\Rightarrow$a² + $\frac{1}{a^2}$ = 11 ( Ans )


ii ) Find value of : a³ - $\frac{1}{a^3}$
Now taking Cube of equation 1 , we get

( a - $\frac{1}{a}$ )³ = 3³

$\Rightarrow$a³ - $\frac{1}{a^3}$ - 3 ( a ) ( $\frac{1}{a}$ ) ( a - $\frac{1}{a}$ ) = 27 [ As we know ( a - b )³ = a³ - b³ - 3ab( a - b ) ]

$\Rightarrow$a³ - $\frac{1}{a^3}$ ​- 3 ( 3 ) = 27 ( Given ​a - $\frac{1}{a}$ = 3 ​ )

$\Rightarrow$a³ - $\frac{1}{a^3}$ - 9 = 27

$\Rightarrow$a³ - $\frac{1}{a^3}$ = 27 + 9

$\Rightarrow$a³ - $\frac{1}{a^3}$ = 36 ( Ans )