# A Lollipop with Roots

$$Given$$:

$$S$$, $$T$$, $$U$$, $$V$$. are distinct digits which can vary from zero to nine, with $$V>U$$.

$$ST$$, $$STT$$ are concatenated Numbers.

Deduce S, T, U, V from the following relationship.

$$ST=\sqrt[U]{STT} \times \sqrt[V]{STT}$$

It works with

$$S=1,T=0,U=3,V=6$$.

We can rewrite the equation as follows:

$$ST=\sqrt[U]{STT}\times\sqrt[V]{STT}=(STT)^{1/U}\times(STT)^{1/V}=(STT)^{\frac{1}{U}+\frac{1}{V}}=(STT)^{\frac{U+V}{U\times V}}$$ $$(ST)^{U*V}=(STT)^{U+V}$$

Clearly this works if

$$ST=10,STT=100$$, and $$U*V=2*(U+V)$$,

which yields the solution stated.

• Great!!!...got it – Uvc Jun 14 at 15:21