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If one root of the equation ax^2 + bx + ...

If one root of the equation `ax^2 + bx + c = 0` is equal to the`n^(th)` power of the other, then `(ac^n)^(1/(n+1)) + (a^n c)^(1/(n+1)) + b` is equal to

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Let `alpha, beta` are roots of `ax^(2)+bx+c=0`
Given, `alpha =beta^(n)`
`impliesalphabeta=c/aimpliesbeta^(n+1)=c/a`
` impliesbeta=((c)/(a))^(1//(n+1))` It must satisfy `ax^(2)+bx+c=0`
`i.e. a ((c)/(a))^(2//(n+1))+b((c)/(a))^(1//(n+1))+c=0`
`implies(a.c^(2//(n+1)))/(a^(2//(n+1)))+(b.c^(1//(n+1)))/(a^(1//(n+1)))+c=0`
`implies(c^(1//(n+1)))/(a^(1//(n+1))){(a.c^(1//(n+1)))/(a^(1(n+1)))+b+(c.a^(1//(n+1)))/(c^(1//(n+1)))}=0`
`impliesa^(n(n+1)_(c)1//(n+1))+b+c^(n//(n+1))a^(1//(n+1))=0`
`implies(a^(n)c)^(1//(n+1))+(c^(n)a)^(1//(n+1))+b=0`
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