Question

A^2+1/A^2=23 find a^3+1a^3​

Answer 1

${a}^{2} + \frac{1}{ {a}^{2} } = 23 \\ ( {a + \frac{1}{a} })^{2} - 2 = 23 \\ ( {a + \frac{1}{a} })^{2} = 25 \\ a + \frac{1}{a} = ±5 \\ if \: a + \frac{1}{a} = 5 \\ {a}^{3} + \frac{1}{ {a}^{3} } = ( {a + \frac{1}{a} })^{3} - 3(a + \frac{1}{a} ) \\ = {5}^{3} - 3 \times 5 \\ = 125 - 15 \\ = 110 \\ if \: a + \frac{1}{a} = - 5 \\ {a}^{3} + \frac{1}{ {a}^{3} } = ( {a + \frac{1}{a} })^{3} - 3(a + \frac{1}{a} ) \\ = ({ - 5})^{3} - 3( - 5) \\ = - 125 + 15 \\ = - 110$