I need help rationalising this

$$\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1) }$$

I'm so stuck with this problem!

  • $\begingroup$ Treat it like a product of four fractions. Rationalize the denominator of each factor, and multiply. $\endgroup$ – D.B. Mar 4 at 3:20

The deepest root is the place to start. Multiply top and bottom by $\sqrt[16]5-1$. You don't have any more sixteenth roots, so this is progress. Keep going.

  • $\begingroup$ Once you remove the sixteenth root others will automatically be removed from the formula $(a+b)(a-b) =a^2 - b^2$ $\endgroup$ – Abhinav Mar 4 at 3:43
  • $\begingroup$ @Abhinav: this step leaves us with some eighth roots. Now we rationalize those and get down to fourth roots. $\endgroup$ – Ross Millikan Mar 4 at 3:45

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