MHB What is the Sum of Two Sixth Powers?

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The discussion revolves around finding the sum of two sixth powers given the equations involving square roots and cube roots of two variables, x and y. By setting variables φ and ψ as the sixth roots of x and y, the equations simplify to φ^3 + ψ^3 = 35 and φ^2 + ψ^2 = 13. The solutions for φ and ψ are determined to be 2 and 3, respectively, leading to φ + ψ = 5. Consequently, the sixth powers of these values yield a final result of x + y = 793. The mathematical approach effectively demonstrates the relationship between the roots and their powers.
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$\sqrt {x}+\sqrt {y}=35$

$\sqrt [3]{x}+\sqrt[3] {y}=13$

find x+y
 
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Re: operation of root equation

Albert said:
$\sqrt {x}+\sqrt {y}=35$

$\sqrt [3]{x}+\sqrt[3] {y}=13$

find x+y

Setting $\displaystyle \varphi = x^{\frac{1}{6}}$ and $\displaystyle \psi = y^{\frac{1}{6}}$ You obtain...

$\displaystyle \varphi^{3} + \psi^ {3} = 35$

$\displaystyle \varphi^{2} + \psi^ {2} = 13$ (1)

... and the (1) has solutions $\displaystyle \varphi=2\ \psi= 3$ and $\displaystyle \varphi=3\ \psi= 2$ so that is in any case $\displaystyle \varphi + \psi = 5$... Kind regards

$\chi$ $\sigma$
 
Re: operation of root equation

Setting $\chi = \varphi^6$ and $\sigma = \psi^6$, we get $\chi + \sigma = \varphi^6 + \psi^6 = 2^6 + 3^6 = 793$.

Kind regards,

$\text{I}\Lambda\Sigma$
 
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