First slide

Theory of expressions

Question

Let $α$ and $β$ be the solutions of the quadratic equation $x^{2} - 1154 x + 1 = 0$ , then the value of $α^{\frac{1}{4}} + β^{\frac{1}{4}}$ is equal to ________.

Moderate

Solution

$α + β = 1154 and αβ = 1$
$(\sqrt{α} + \sqrt{β})^{2} = α + β + 2 \sqrt{αβ} = 1154 + 2 = 1156 = (34)^{2}$
$\Rightarrow \sqrt{α} + \sqrt{β} = 34$
Again ${(α^{1 / 4} + β^{1 / 4})}^{2} = \sqrt{α} + \sqrt{β} + 2 (αβ)^{1 / 4} = 34 + 2 = 36$
$α^{1 / 4} + β^{1 / 4} = 6$

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