Let f be $f:[a,b] \rightarrow \mathscr{R}$. If $f$ is a fuction of bounded variation then $|f|^p$ is also a function of bounded variation, for any $1\leq p < \infty$
I'll try to write what I did. Suppose that f is a function of bounded variation. Then there is $M>0$ such that for every partition $P=\{x_{0},x_{1},...,x_{n}\}$. We have $\sum_{k=1}^{n}|{f(x_{k})-f(x_{k-1})|\leq M}$. Then $\sum_{k=1}^{n}||{f(x_{k})|^p-|f(x_{k-1})|^p|\leq \sum_{k=1}^{n}|{(f(x_{k}))^p-(f(x_{k-1}))^p|}}$
That's all I've done.
Use MVT to show that $||x|^{p}-|y|^{p}| \leq p M^{p-1} |x-y|$ where $M =\max \{|a|,|b|\}$. From this your result follows by definition. [Replace $x$ by $f(x)$ and $y$ by $f(y)$ in this inequality].