Question

If a=6+2root3, then find the value of a-1/a

Answer 1

hope this helps .........

Answer 2

Answer:

$a-\frac{1}{a}=\frac{69+25\sqrt3}{12}$

Step-by-step explanation:

Given : $a=6+2\sqrt3$

To find : The value of $a-\frac{1}{a}$ ?

Solution :

$a=6+2\sqrt3$

First we find $\frac{1}{a}$

i.e. $\frac{1}{a}=\frac{1}{6+2\sqrt3}$

Rationalize the denominator,

$\frac{1}{a}=\frac{1}{6+2\sqrt3}\times \frac{6-2\sqrt3}{6-2\sqrt3}$

$\frac{1}{a}=\frac{6-2\sqrt3}{6^2-(2\sqrt3)^2}$

$\frac{1}{a}=\frac{6-2\sqrt3}{36-12}$

$\frac{1}{a}=\frac{6-2\sqrt3}{24}$

$\frac{1}{a}=\frac{3-\sqrt3}{12}$

Substitute back in expression,

$a-\frac{1}{a}=6+2\sqrt3-\frac{3-\sqrt3}{12}$

$a-\frac{1}{a}=\frac{12(6+2\sqrt3)-(3-\sqrt3)}{12}$

$a-\frac{1}{a}=\frac{72+24\sqrt3-3+\sqrt3}{12}$

$a-\frac{1}{a}=\frac{69+25\sqrt3}{12}$

Therefore, $a-\frac{1}{a}=\frac{69+25\sqrt3}{12}$