Question

a is not equal to 0 and a-1/a=4 find a^2+1/a^2 and a^4- 1/a^4 and a^3 - 1/a^3
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Answer 1

$a - \frac{1}{a} = 4 \\ \\ ( {a - \frac{1}{a} })^{2} = 16 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 16 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 18 ...(1)\\ \\ ( { {a}^{2} + \frac{1}{ {a}^{2} } })^{2} = 324 \\ \\ ( { {a}^{2} - \frac{1}{ {a}^{2} } })^{2} + 4 = 324 \\ \\ ( { {a}^{2} - \frac{1}{ {a}^{2} } })^{2} = 320 \\ \\ {a}^{2} - \frac{1}{ {a}^{2} } = 8 \sqrt{5} ...(2) \\ \\ now \: \: {a}^{4} - \frac{1}{ {a}^{4} } = ( {a}^{2} + \frac{1}{ {a}^{2} } )( {a}^{2} - \frac{1}{ {a}^{2} } ) \\ = 18 \times 8 \sqrt{5} \\ = 144 \sqrt{5} \\ \\ \\ {a}^{3} - \frac{1}{ {a}^{3} } = ( {a - \frac{1}{a} })^{3} + 3(a - \frac{1}{a} ) \\ = {4}^{3} + 3 \times 4 \\ = 64 + 12 \\ = 76$