If α, βbe the roots of the equation x^2-p x+q=0 and α>0, β>0, then the value of α^1 / 4+β^1 / 4

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If α, βbe the roots of the equation x^2-p x+q=0 and α>0,...

If $\alpha, \beta$ be the roots of the equation $x^{2}-p x+q=0$ and $\alpha>0, \beta>0$, then the value of $\alpha^{1 / 4}+\beta^{1 / 4}$ is $\left(p+6 \sqrt{q}+4 q^{1 / 4} \sqrt{p+2 \sqrt{q}}\right)^{k}$, where $k$ is equal to (A) 1 (B) $\frac{1}{2}$ (C) $\frac{1}{3}$ (D) $\frac{1}{4}$

If $\alpha, \beta$ be the roots of the equation $x^{2}-p x+q=0$ and $\alpha>0, \beta>0$, then the value of $\alpha^{1 / 4}+\beta^{1 / 4}$ is $\left(p+6 \sqrt{q}+4 q^{1 / 4} \sqrt{p+2 \sqrt{q}}\right)^{k}$, where $k$ is equal to
(A) 1
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{1}{4}$

Key Concepts

Algebraic Identities and Manipulation

Algebraic identities such as the binomial expansion and other formulas for sums and powers are often used in problems dealing with expressions involving radicals and powers. They facilitate the rewriting of complex expressions in a form that makes comparison or further simplification possible, particularly when identifying exponents or parameters (like k) that equate two seemingly different expressions.

Exponent Equivalence and Expression Matching

Exponent equivalence involves comparing two expressions by rewriting them in a common form to identify an unknown exponent or parameter. In the context of the problem, the goal is to express a sum of radical terms as a single expression raised to a power. This requires matching the given complex expression to a desired form by equating corresponding parts and extracting the value of the exponent k.

Vieta's Formulas

Vieta's formulas establish the relationships between the coefficients of a polynomial and the sums and products of its roots. In a quadratic equation of the form x² - px + q = 0, these formulas tell us that the sum of the roots is equal to p (? + ? = p) and the product of the roots is equal to q (?? = q). This connection is vital for expressing functions of the roots in terms of the coefficients of the equation.

Radical Expressions with Fractional Exponents

Radical expressions involving fractional exponents, such as ?^(1/4) and ?^(1/4), can be manipulated using the properties of surds and exponents. Understanding how to simplify and combine such expressions is key to solving problems where the root expressions are transformed, for example, when summing terms like ?^(1/4) + ?^(1/4) and relating them to other algebraic forms involving square roots or fourth roots.

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