Prove the following
Question:

If $\alpha$ and $\beta$ be two roots of the equation $x^{2}-64 x+256=0$.

Then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is:

  1. (1) 2

  2. (2) 3

  3. (3) 1

  4. (4) 4


Correct Option: 1

Solution:

$\because \alpha+\beta=64, \alpha \beta=256$

$\frac{\alpha^{3 / 8}}{\beta^{5 / 8}}+\frac{\beta^{3 / 8}}{\alpha^{5 / 8}}=\frac{\alpha+\beta}{(\alpha \beta)^{5 / 8}}=\frac{64}{\left(2^{8}\right)^{5 / 8}}=\frac{64}{32}=2$

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