Let's simplify the expression \((15a^0 \cdot b^2 \cdot c^{34})(3a^{16} \cdot b^{-29} \cdot c^0)\).
Step 1: Simplify \(a^0\) and \(c^0\) \[ a^0 = 1 \quad \text{and} \quad c^0 = 1 \] Thus, \(15a^0 = 15\) and \(c^0 = 1\).
Step 2: Combine the constants \[ 15 \cdot 3 = 45 \]
Step 3: Combine the \(a\) terms \[ a^0 \cdot a^{16} = a^{0 + 16} = a^{16} \]
Step 4: Combine the \(b\) terms \[ b^2 \cdot b^{-29} = b^{2 - 29} = b^{-27} \]
Step 5: Combine the \(c\) terms \[ c^{34} \cdot c^0 = c^{34 + 0} = c^{34} \]
Step 6: Combine all these results together: \[ (15a^0 \cdot b^2 \cdot c^{34})(3a^{16} \cdot b^{-29} \cdot c^0) = 45a^{16} \cdot b^{-27} \cdot c^{34} \]
Step 7: Express the final format: \[ = \frac{45a^{16} c^{34}}{b^{27}} \]
Now, we can match this to the given options. The simplified expression is: \[ \frac{45a^{16} c^{34}}{b^{27}} \]
Thus, the correct answer is:
D: \(\frac{45a^{16} c^{34}}{b^{27}}\)