Question

If x3+1/x3=18 then what will be the value of x4+1/x4?

Answer 1

${(x + \frac{1}{x}) }^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3(x + \frac{1}{x} ) \\ \\ let \: x + \frac{1}{x} = y \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = {y}^{3} - 3y \\ \\ {y}^{3} - 3y = 18 \\ \\ y( {y}^{2} - 3) = 3 \times 6 \\ \\ y = 3 \\ \\ {y}^{2} - 3 = 6 \\ \\ now \\ \\ {y}^{4} = {x}^{4} + \frac{1}{ {x}^{4} } + 4( {x}^{2} + \frac{1}{ {x}^{2} } ) + 6 \\ \\ {y}^{4} = {x}^{4} + \frac{1}{ {x}^{4} } + 4( {x}^{2} + \frac{1}{ {x}^{2} } + 2 - 2) + 6 \\ \\ {y}^{4} = {x}^{4} + \frac{1}{ {x}^{4} } + 4( {y}^{2} - 2 ) + 6 \\ \\ {y}^{4} = {x}^{4} + \frac{1}{ {x}^{4} } + 4 {y}^{2} - 2 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = {y}^{4} - 4 {y}^{2} + 2 \\ \\ = {3}^{4} - 4 \times {3}^{2} + 2 \\ \\ = 47 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 47$

hope it helps