Question

plz solve it.........​

Answer 1

Given : x = $9-4\sqrt{5}$

Then, $\frac{1}{x}=\frac{1}{9-4\sqrt{5}}$

$=\frac{(9+4\sqrt{5})}{(9-4\sqrt{5})(9+4\sqrt{5})}$

$=\frac{9+4\sqrt{5}}{81-80}$

$=9+4\sqrt{5}$

Now, $x^{2}-\frac{1}{x^{2}}$

$=(x+\frac{1}{x})(x-\frac{1}{x})$

$\small{=(9-4\sqrt{5}+9+\sqrt{5})(9-4\sqrt{5}-9-4\sqrt{5})}$

$=18\times (-8\sqrt{5})$

$=-144\sqrt{5}$

Again, $x^{3}-\frac{1}{x^{3}}$

$=(x-\frac{1}{x})^{3}+3.x.\frac{1}{x}(x-\frac{1}{x})$

$\small{=(9-4\sqrt{5}-9-4\sqrt{5})^{3}+3(9-4\sqrt{5})}$

$=(-8\sqrt{5})^{3}+27-12\sqrt{5}$

$=-2560\sqrt{5}+27-12\sqrt{5}$

$=27-2572\sqrt{5}$