As we know that
Dimension of energy $=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
Let us consider $\mathrm{M}_1, \mathrm{~L}_1, \mathrm{~T}_1$ and $\mathrm{M}_2, \mathrm{~L}_2, \mathrm{~T}_2$ are units of mass, length and time in given two systems.
$\mathrm{M}_1=1 \mathrm{~kg}, \mathrm{~L}_1=1 \mathrm{~m}, \mathrm{~T}_1=1 \mathrm{~s}, \mathrm{n}_1=$ SI system unit $=5 \mathrm{~J}$
$\mathrm{M}_2=\alpha \mathrm{kg}, \mathrm{L}_2=\beta \mathrm{m}, \mathrm{T}_2=\gamma \mathrm{s}, \mathrm{n}_2=$ New system unit $=$ ?
The magnitude of a physical quantity remains the same, whatever be the system of units of its measurement i.e.,
$$
n_1 u_1=n_2 u_2
$$
So, $n_2=n_1 \frac{u_1}{u_2}=n_1 \frac{\left[\mathrm{M}_1 \mathrm{~L}_1^2 \mathrm{~T}_1^{-2}\right]}{\left[\mathrm{M}_2 \mathrm{~L}_2^2 \mathrm{~T}_2^{-2}\right]}$
$$
\begin{aligned}
&=5\left[\frac{\mathrm{M}_1}{\mathrm{M}_2}\right] \times\left[\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right]^2 \times\left[\frac{\mathrm{T}_1}{\mathrm{~T}_2}\right]^{-2} \\
&=5\left[\frac{1}{\alpha} \mathrm{kg}\right] \times\left[\frac{1}{\beta} \mathrm{m}\right]^2 \times\left[\frac{1}{\gamma} \mathrm{s}\right]^{-2} \\
&=5 \times \frac{1}{\alpha} \times \frac{1}{\beta^2} \times \frac{1}{\gamma^{-2}}=\frac{5 \gamma^2}{\alpha \beta^2}
\end{aligned}
$$
$(\because 5$ is constant so neglect it)
So, new system unit of energy will be $\frac{\gamma^2}{\alpha \beta^2}$, or $\left[\alpha^{-1} \beta^{-2} \gamma^2\right]$.