If α and β are the roots of the equation 8 x 2 - 3 x + 27 = 0 , then the value of α 2 β 1 3 + β 2 α 1 3 is

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Question: Answered & Verified by Expert

If

α

and

β

are the roots of the equation

8 x^{2} - 3 x + 27 = 0,

then the value of

{(\frac{α^{2}}{β})}^{\frac{1}{3}} + {(\frac{β^{2}}{α})}^{\frac{1}{3}}

is

MathematicsQuadratic EquationJEE Main

Options:

A $\frac{1}{3}$
B $\frac{1}{4}$
C $\frac{1}{5}$
D $\frac{1}{6}$

Solution:

2678 Upvotes Verified Answer

The correct answer is:

\frac{1}{4}

α + β = \frac{3}{8}, α β = \frac{27}{8}

{(\frac{α^{2}}{β})}^{\frac{1}{3}} + {(\frac{β^{2}}{α})}^{\frac{1}{3}} = \frac{{(α^{3})}^{\frac{1}{3}} + {(β^{3})}^{\frac{1}{3}}}{{(α β)}^{\frac{1}{3}}}

\frac{α + β}{{(α β)}^{\frac{1}{3}}} = \frac{\frac{3}{8}}{{(\frac{27}{8})}^{\frac{1}{3}}} = \frac{\frac{3}{8}}{\frac{3}{2}} = \frac{1}{4}

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