# Number Properties Puzzle – Challenge 19

Enjoy working with numbers and solving puzzles? Test your mind with this tough math challenge. Let’s challenge your brain!

## Challenge:

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

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The correct answer is $$z^{\frac{-4}{3}}$$.

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
$$p^{\frac{1}{6}} = z^{\frac{-1}{2}} →(p^{\frac{1}{6}} )^6 = (z^{\frac{-1}{2}})^6 → p =z^{-3}→$$
$$(p)^{\frac{-4}{9}} = (z^{-3)}) ^{\frac{-4}{9}} = z^{\frac{-12}{9}} = z^{\frac{-4}{3}}$$
$$t = p^{\frac{-4}{9}}$$ and $$(p)^{\frac{-4}{9}} = z^{\frac{-4}{3}}$$, therefore, $$t =z^{\frac{-4}{3}}$$