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Without the use of a calculator, and showing your work, simplify:

\(\displaystyle \frac{1}{2}\left(\left(239+169\sqrt{2}\right)^{ \frac{1}{7}}-\left(29\sqrt{2}-41\right)^{ \frac{1}{5}}\right)\)

edit: My apologies...I was careless in my first statement of the problem...

\(\displaystyle \frac{1}{2}\left(\left(239+169\sqrt{2}\right)^{ \frac{1}{7}}-\left(29\sqrt{2}-41\right)^{ \frac{1}{5}}\right)\)

edit: My apologies...I was careless in my first statement of the problem...

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