The product of divisors of $\text{N}$ is

$N={p_1}^{n_1}\times {p_2}^{n_2}\times {p_3}^{n_3}\times \mathrm{.....}{p_n}^{n_k}$

The divisors of ${p_1}^{n_1}=1,p_1,{p_1}^2,{p_1}^3,\mathrm{......},{p_1}^{n_1}$

The divisors of ${p_2}^{n_2}=1,p_2,{p_2}^2,{p_2}^3,\mathrm{......},{p_2}^{n_2}$

The divisors of ${p_3}^{n3}=1,p_3,{p_3}^2,{p_3}^3,\mathrm{......},{p_3}^{n_3}$

The product of divisors of $N$

$(1\times p_1\times {p_1}^2\times \mathrm{......}\times {p_1}^{n_1})(1\times p_2\times {p_2}^2\times \mathrm{......}\times {p_2}^{n_2})(1\times p_3\times {p_3}^2\times \mathrm{......}\times {p_3}^{n_3})\mathrm{...}$

$({p_1}^0\times {p_1}^1\times {p_1}^2\times \mathrm{....}\times {p_1}^{n_1})({p_2}^0\times {p_2}^1\times {p_2}^2\times \mathrm{....}\times {p_2}^{n_2})({p_3}^0\times p_3\times {p_3}^2\times \mathrm{....}\times {p_3}^{n_3})\mathrm{....}$ $=\left({p_1}^{0+1+2+\mathrm{......}+n_1}\right)\left({p_2}^{0+1+2+\mathrm{......}+n_2}\right)\left({p_3}^{0+1+2+\mathrm{......}+n_3}\right)\mathrm{...}$

$=\left({p_1}^{\frac{n_1(n_1+1)}{2}}\right)\left({p_2}^{\frac{n_2(n_2+1)}{2}}\right)\left({p_3}^{\frac{n_3(n_3+1)}{2}}\right)\mathrm{.....}$

The number of divisors of $n$ is $n+1$

$=\left({p_1}^{\frac{n_{1}d\left(N\right)}{2}}\right)\left({p_2}^{\frac{n_{2}d\left(N\right)}{2}}\right)\left({p_3}^{\frac{n_{3}d\left(N\right)}{2}}\right)\mathrm{....}$

$={\left({p_1}^{n_1}\right)}^{\frac{d\left(N\right)}{2}}{\left({p_2}^{n_2}\right)}^{\frac{d\left(N\right)}{2}}{\left({p_3}^{n_3}\right)}^{\frac{d\left(N\right)}{2}}\mathrm{....}$

$={({p_1}^{n_1}\times {p_2}^{n_2}\times {p_3}^{n_3}\mathrm{....})}^{{ }^{\frac{d\left(N\right)}{2}}}$

$=N^{\frac{d\left(N\right)}{2}}$