Question

se știe că
$\frac{1}{a} - a = 8$
aflați:


a)
${a}^{2} + \frac{1}{a {}^{2} }$
b)
${a}^{4} + \frac{1}{a {}^{4} }$

​

Answer 1

$\frac{1}{a} - a = 8 \\ ( { \frac{1}{a} - a })^{2} = \frac{1}{ {a}^{2} } - 2 \times \frac{1}{a} \times a + {a}^{2} = {8}^{2} \\ = \frac{1}{ {a}^{2} } - 2 + {a}^{2} = 64 \\ \frac{1}{ {a}^{2} } + {a}^{2} = 66 \\ ( { \frac{1}{ {a}^{2} } + {a}^{2} })^{2} = \frac{1}{ {a}^{4} } + 2 \times \frac{1}{ {a}^{2} } \times {a}^{2} + {a}^{4} = {66}^{2} \\ \frac{1}{ {a}^{4} } + {a}^{4} = 4356 - 2 \\ \frac{1}{ {a}^{4} } + {a}^{4} = 4354$