Question

Expand $\mathbf{6}\sqrt{\mathbf{\prod }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{3}}\mathbf{\prod }_{\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{\text{'}}}^{\mathbf{2}\mathbf{\text{'}}}{\mathbf{D}}_{\mathbf{km}}}$.

Short answer

Therefore, the expanded expression is $6\sqrt{\left({\mathrm{D}}_{11\text{'}}{\mathrm{D}}_{12\text{'}}\right)\left({\mathrm{D}}_{21\text{'}}{\mathrm{D}}_{22\text{'}}\right)\left({\mathrm{D}}_{31\text{'}}{\mathrm{D}}_{32\text{'}}\right)}$.

Step-by-step solution

Determine the expansion of the term k and m .

Consider the given expression as

$\mathbf{6}\sqrt{\mathbf{\prod }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{3}}\mathbf{\prod }_{\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{\text{'}}}^{\mathbf{2}\mathbf{\text{'}}}{\mathbf{D}}_{\mathbf{km}}}$

From the above expression, it can be seen that the value of k is varies from 1 to 3 and m varies from 1 to 2 which results the 3 and 2 terms respectively.

determine the expanded expression.

Expand the expression for k .

$6\sqrt{\prod _{1\text{'}}^{2\text{'}}\left(D_{1m}{D}_{2m}{D}_{3m}\right)}$

Expand the expression for m .

$$\begin{gathered} 6\sqrt{\left(D_{11\text{'}}{D}_{21\text{'}}{D}_{31\text{'}}\right)\left(D_{12\text{'}}{D}_{22\text{'}}{D}_{32\text{'}}\right)} \\ 6\sqrt{\left(D_{11\text{'}}{D}_{12\text{'}}\right)\left(D_{21\text{'}}{D}_{22\text{'}}\right)\left(D_{31\text{'}}{D}_{32\text{'}}\right)} \end{gathered}$$

Hence, the expanded form is $6\sqrt{\left(D_{11\text{'}}{D}_{12\text{'}}\right)\left(D_{21\text{'}}{D}_{22\text{'}}\right)\left(D_{31\text{'}}{D}_{32\text{'}}\right)}$.