b) \( \left(\frac{1}{9} m^{5}\right)^{2}\left(\frac{1}{4} m^{7}\right)^{3} \)

To simplify the expression: \[ \left(\frac{1}{9} m^{5}\right)^{2}\left(\frac{1}{4} m^{7}\right)^{3} \] **Step 1: Apply the Power to Each Factor** First, apply the exponent to each part inside the parentheses: \[ \left(\frac{1}{9}\right)^{2} \cdot (m^{5})^{2} \cdot \left(\frac{1}{4}\right)^{3} \cdot (m^{7})^{3} \] **Step 2: Simplify Each Part** Calculate each part separately: \[ \left(\frac{1}{9}\right)^{2} = \frac{1}{81} \] \[ (m^{5})^{2} = m^{10} \] \[ \left(\frac{1}{4}\right)^{3} = \frac{1}{64} \] \[ (m^{7})^{3} = m^{21} \] **Step 3: Multiply the Simplified Parts Together** Now, multiply all the simplified parts: \[ \frac{1}{81} \cdot m^{10} \cdot \frac{1}{64} \cdot m^{21} = \frac{1}{81 \times 64} \cdot m^{10+21} \] **Step 4: Final Simplification** Calculate the numerical part and combine the exponents: \[ 81 \times 64 = 5184 \] \[ m^{10+21} = m^{31} \] So, the simplified expression is: \[ \frac{m^{31}}{5184} \] **Final Answer:** \[ \boxed{\dfrac{m^{31}}{5184}} \]