Number Properties Puzzle – Challenge 19

Challenge:

If \(t^{\frac{1}{2}} = p^{\frac{-2}{3}}\) and \(p^{\frac{1}{2}} = z^{\frac{-3}{5}}\), what is the value of t in terms of z?

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To find t in terms of z, first we need to find the value of t in terms of p:
\(t^{\frac{1}{2}} = p^{\frac{-2}{3}} → (t^{\frac{1}{2}})^2= (p^{\frac{-2}{3}})^2 → t = p^{\frac{4}{9}}\)
Now, solve for p
\(p^{\frac{1}{2}} = z^{\frac{-3}{5}} →(p^{\frac{1}{2}} )^2 = (z^{\frac{-3}{5}})^2 → p =z^{\frac{9}{25}}→\)
\(p^{\frac{4}{9}} = z^{\frac{4}{25}} →\)
\(t = p^{\frac{4}{9}}\) and \((p)^{\frac{4}{9}} = z^{\frac{4}{25}}\), therefore, \(t =z^{\frac{4}{25}}\)