How to find radix?

Question

$\sqrt{(144)_r} =(12)_r$

I have tried $\sqrt{r^2+4r+4} =r+2$

From this, I am unable to find the value of $r$. Can anyone help me out to solve this problem?

Answer 1

You are on the right track, proceed as follows $$\begin{align*} \sqrt{(144)_r}&=\sqrt{r^2+4r+4} \\ &= \sqrt{(r+2)^2} \\&= (r+2) \\&=(12)_r \end{align*}$$ $LHS=RHS$

The equation is true for any $r,$ but base $r$ in $(144)_r$ with a digit of $4$, so $r>4$