Question

if x²+ 1/x²=34 find x³+1/x³

Answer 1

x² + 1/x² = 34

Add 2(x)(1/x) on both sides

${x}^{2} + \frac{1}{ {x}^{2} } + 2(x)( \frac{1}{x} ) = 34 + 2(x)( \frac{1}{x})$
$= > (x + \frac{1}{x} ) {}^{2} = 34 + 2 \\ \\ = > (x + \frac{1}{x} ) {}^{2} = 36 \\ \\ = > x + \frac{1}{x} = \sqrt{36} \\ \\ = > x + \frac{1}{x} = 6$


We know that


${x}^{3} + \frac{1}{ {x}^{3} } = (x + \frac{1}{x} )( {x}^{2} + \frac{1}{ {x}^{2} } - (x)( \frac{1}{x} ))$


$= > {x}^{3} + \frac{1}{ {x}^{3} } = (6)(34 - 1) \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } = 6 \times 33 \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } = 198$