We need to check that which of the given options is/are equal to $\(\mathrm{q}^{21}\)$ for all positive values of q .

(A). $\(\left(\mathrm{q}^3\right)^9 \cdot\)$

As $\(\left(\mathrm{x}^{\mathrm{a\right)^b=\mathrm{x}^{\mathrm{ab}\)$

we get $\(\left(\mathrm{q}^3\right)^9=\mathrm{q}^{3 \times 9}=\mathrm{q}^{27}\)$

(B). $\(q^{13}+q^{14}=q^{13}(1+q)[m]$ which is not equal to $[m]q^{27}\)$

(C). $\(q^9+q^9+q^9=3 q^9\)$ which is also not equal to $\(q^{27}\)$

(D). $\(\left(\mathrm{q}^9\right) \times\left(\mathrm{q}^9\right) \times\left(\mathrm{q}^9\right)=\mathrm{q}^{9+9+9}=\mathrm{q}^{27}\)$

$ \((\because\)\( x^a \times x^b=x^{ab})\)$

Hence only options (A) \& (D) are equal to $\(\mathrm{q}^{27}\)$