MHB What is the Sum of Two Sixth Powers?

[Albert1]

$\sqrt {x}+\sqrt {y}=35$

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

find x+y

 

[chisigma]

Re: operation of root equation

$\sqrt {x}+\sqrt {y}=35$

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

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

$\chi$ $\sigma$

 

[I like Serena]

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,

$\text{I}\Lambda\Sigma$