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Question Number 110136 by qwerty111 last updated on 27/Aug/20

	Answered by $@y@m last updated on 27/Aug/20

	$L e t 2020^{m} = a$ $G i v e n,$ $a + \frac{1}{a} = 3 . . . . . (1)$ $S q u a r r i n g b o t h s i d e s,$ $\Rightarrow a^{2} + \frac{1}{a^{2}} + 2 = 9$ $\Rightarrow a^{2} + \frac{1}{a^{2}} = 7$ $\Rightarrow a^{2} + \frac{1}{a^{2}} - 2 = 5$ $\Rightarrow a - \frac{1}{a} = \sqrt{5} . . . (2)$ $C u b i n g b o t h s i d e s,$ $\Rightarrow {(a - \frac{1}{a})}^{3} = \sqrt{5}^{3}$ $\Rightarrow a^{3} - \frac{1}{a^{3}} - 3 \sqrt{5} = 5 \sqrt{5}$ $\Rightarrow a^{3} - \frac{1}{a^{3}} = 8 \sqrt{5} . . . (3)$ $C u b i m g b o t h s i d e s o f (1),$ $a^{3} + \frac{1}{a^{3}} + 3 \times 3 = 27$ $a^{3} + \frac{1}{a^{3}} = 18 . . . (4)$ $b y (3) \times (4),$ $a^{6} - \frac{1}{a^{6}} = 144 \sqrt{5} . . . . (5)$ $B y (5) \div (2)$ $\frac{a^{6} - \frac{1}{a^{6}}}{a - \frac{1}{a}} = 144$ $∴ \sqrt{\frac{a^{6} - \frac{1}{a^{6}}}{a - \frac{1}{a}}} = 12$
	Answered by Sarah85 last updated on 27/Aug/20

	$x^{m} + x^{- m} = 3$ $x^{2 m} + 1 = 3 x^{m}$ $x^{2 m} - 3 x^{m} + 1 = 0$ $x^{m} = \frac{3 \pm \sqrt{5}}{2}$ $\Rightarrow x^{6 m} = {(\frac{3 \pm \sqrt{5}}{2})}^{6} = 161 \pm 72 \sqrt{5}$ $x^{6 m} - x^{- 6 m} = \pm 144 \sqrt{5}$ $x^{m} - x^{- m} = \pm \sqrt{5}$ $\Rightarrow answer is 12 independent of x$
	Answered by Rasheed.Sindhi last updated on 27/Aug/20

	$2020 = a$ $a^{m} + a^{- m} = 3, \sqrt{\frac{a^{6 m} - a^{- 6 m}}{a^{m} - a^{- m}}} = ?$ $a^{2 m} + a^{- 2 m} + 2 = 9$ $a^{2 m} + a^{- 2 m} = 7$ $a^{4 m} + a^{- 4 m} = 47$ $\sqrt{\frac{a^{6 m} - a^{- 6 m}}{a^{m} - a^{- m}}} =$ $\sqrt{\frac{(a^{m} - a^{- m}) (a^{m} + a^{- m}) (a^{4 m} + a^{- 4 m} + 1)}{a^{m} - a^{- m}}}$ $\sqrt{(a^{m} + a^{- m}) (a^{4 m} + a^{- 4 m} + 1)}$ $\sqrt{(3) (47 + 1)}$ $\sqrt{(3) (48)}$ $12 f o r a n y v a l u e o f a$
	Answered by Rasheed.Sindhi last updated on 27/Aug/20

	$2020 = a$ $a^{m} + a^{- m} = 3, \sqrt{\frac{a^{6 m} - a^{- 6 m}}{a^{m} - a^{- m}}}$ $a^{2 m} + a^{- 2 m} = 7$ $a^{4 m} + a^{- 4 m} = 47$ $= \sqrt{\frac{a^{6 m} - a^{- 6 m}}{a^{m} - a^{- m}} \times \frac{a^{m} + a^{- m}}{a^{m} + a^{- m}}}$ $= \sqrt{\frac{3 (a^{6 m} - a^{- 6 m})}{a^{2 m} - a^{- 2 m}}}$ $= \sqrt{\frac{3 (a^{2 m} - a^{- 2 m}) (a^{4 m} + a^{- 4 m} + 1)}{a^{2 m} - a^{- 2 m}}}$ $= \sqrt{3 (a^{4 m} + a^{- 4 m} + 1)}$ $= \sqrt{3 (47 + 1)} = 12$
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