Question

23.) if a-(1/a)= 3, find the value of a²- (1/ a²)​

Answer 1

1. Squaring on both sides (a-1/a)²=(3)²
2. a²-1/a2+2=9
3. a²-1/a2=9-2
4. a²-1/a2 =7 Answer

Answer 2

Given,

$\[a - \frac{1}{a} = 3\]$

$\[{a^2} - \left( {\frac{1}{{{a^2}}}} \right) = ?\]$

Solution,

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

Squaring on both side of given condition,

$\[{\left( {a - \frac{1}{a}} \right)^2} = {3^2}\]$

$\[{a^2} + \frac{1}{{{a^2}}} - 2 = 9\]$

$\[{a^2} + \frac{1}{{{a^2}}} = 11\]$-------(2)

Know that,

$\[{\left( {a + \frac{1}{a}} \right)^2} = {a^2} + \frac{1}{{{a^2}}} + 2\]$

Put value from equation 2,

$\[ \Rightarrow {\left( {a + \frac{1}{a}} \right)^2} = 11 + 2\]$

$\[ \Rightarrow \left( {a + \frac{1}{a}} \right) = \sqrt {13} \]$---------------(3)

Multiply equation 1 and 3,

$\[ = \left( {a - \frac{1}{a}} \right)\left( {a + \frac{1}{a}} \right)\]$

$\[ = \left( {{a^2} - \frac{1}{{{a^2}}}} \right)\]$

$\[\begin{array}{l} = 3 \times \sqrt {13} \\ = 3\sqrt {13} \end{array}\]$

Hence the value of $\[\left( {{a^2} - \frac{1}{{{a^2}}}} \right)\]$ is $\[3\sqrt {13} \]$.