  A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$,
where $u$ is a word and $k$ is a positive integer and a square is a word of the
form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct
squares in a word is bounded by the length of the word. This conjecture was
proven recently by Brlek and Li. Besides, there exists a stronger upper bound
for binary words conjectured by Jonoska, Manea and Seki stating that for a word
of length $n$ over the alphabet $\left\{a, b\right\}$, if we let $k$ be the
least of the number of a's and the number of b's and $k \geq 2$, then the
number of distinct squares is upper bounded by $\frac{2k-1}{2k+2}n$. In this
article, we prove this conjecture by giving a stronger statement on the number
of distinct powers in a binary word.
