Question

If x^2+1/x^2=23, find x^3+1/x^3,when x>0​

Answer 1

$x^{2} +\frac{1}{x^{2} } = 23\\\\(x+\frac{1}{x} )^{2} =x^{2}+\frac{1}{x^{2} } +2 =23+2 = 25\\\\x+\frac{1}{x} = 5 \\\\(x+\frac{1}{x} )^{3} = x^{3} +\frac{1}{x^{3} } +3*x*\frac{1}{x}(x+\frac{1}{x} ) (I used the identity (x+y)^{3}=x^{3} +y^{3} +3xy(x+y) )\\\\5^{3} =x^{3} +\frac{1}{x^{3} }+3*5\\\\125=x^{3} +\frac{1}{x^{3} }+15\\\\x^{3} +\frac{1}{x^{3} }=125-15 =110$

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