Let - - be two roots of the equation x2-20 14x-5 12 - 0- Then - 8 - - 8 is equal to

X^2+(20)^ 14x+(5)^ 12 = 0 ∴α + β = (20)^ 14, α . β = nbsp;(5)^ 12 α ^8 + β ^8 = (α ^4 + β ^4)^2 2 α ^4 β ^4 ={(α ^2 + β ^2)^2 2 α ^2β ^2}^2 2 α ^4 β ^4 ...

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Standard XIII

Mathematics

Algebra of Roots of Quadratic Equations

Let α, β be t...

Question

Let $α, β$ be two roots of the equation $x^{2} + (20)^{\frac{1}{4}} x + (5)^{\frac{1}{2}} = 0$ . Then $α^{8} + β^{8}$ is equal to

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160

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100

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Solution

The correct option is D 50
$x^{2} + (20)^{\frac{1}{4}} x + (5)^{\frac{1}{2}} = 0$
$∴ α + β = - (20)^{\frac{1}{4}}, α . β = (5)^{\frac{1}{2}}$
$α^{8} + β^{8} = (α^{4} + β^{4})^{2} - 2 α^{4} β^{4}$
$= {(α^{2} + β^{2})^{2} - 2 α^{2} β^{2}}^{2} - 2 α^{4} β^{4}$
$= {[{(α + β)^{2} - 2 α β}^{2} - 2 α^{2} β^{2}]}^{2} - 2 α^{4} β^{4}$
$= {⎡ ⎢ ⎢ ⎣ {⎧ ⎪ ⎨ ⎪ ⎩ 20^{\frac{1}{2}} - {2.5}^{\frac{1}{2}} ⎫ ⎪ ⎬ ⎪ ⎭}^{2} - 2.5 ⎤ ⎥ ⎥ ⎦}^{2} - {2.5}^{2}$
$= (0 - 10)^{2} - 50$
$= 50$

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Let α,β be two roots of the equation x2+(20)14x+(5)12=0. Then α8+β8 is equal to

The correct option is D 50x2+(20)14x+(5)12=0 ∴α+β=−(20)14,α.β=(5)12 α8+β8=(α4+β4)2−2α4β4 ={(α2+β2)2−2α2β2}2−2α4β4 =[{(α+β)2−2αβ}2−2α2β2]2−2α4β4 =⎡⎢ ⎢⎣⎧⎪⎨⎪⎩2012−2.512⎫⎪⎬⎪⎭2−2.5⎤⎥ ⎥⎦2−2.52 =(0−10)2−50 =50

Let $α, β$ be two roots of the equation $x^{2} + (20)^{\frac{1}{4}} x + (5)^{\frac{1}{2}} = 0$ . Then $α^{8} + β^{8}$ is equal to