Question

If a-1/a=2, show that at a⁸+1/a⁴= 34​

Answer 1

Answer:

$a - \frac{1}{a} = 2 \\ \\ (a - \frac{1}{a} ) {}^{2} = 4 \\ \\ a {}^{2} + \frac{1}{a {}^{2} } - 2a. \frac{1}{a} = 4 \\ \\ a {}^{2} + \frac{1}{a {}^{2} } = 4 + 2 \\ \: \: \: \: = 6 \\ \\ (a {}^{2} + \frac{1}{a {}^{2} } ) {}^{2} = 36 \\ \\ a {}^{4} + \frac{1}{a {}^{4} } + 2a {}^{2} . \frac{1}{a {}^{2} } = 36 \\ \\ a {}^{4} + \frac{1}{a {}^{4} } = 36 - 2 \\ \\ \: \: \: \: = 34 \: \: \: (proved) \\ \\ formula - \\ \\ (x - y) {}^{2} = x {}^{2} + y {}^{2} - 2xy \\ \\ (x + y) {}^{2} = x {}^{2} + y {}^{2} + 2xy$