Question

35 Which expression is equivalent to (15a ^ 0 * b ^ 2 * c ^ 34)(3a ^ 16 * b ^ - 29 * c ^ 0) for all values of a, b, and c where the expression is defined?
Q
C (18a ^ 16 * c ^ 34)/(b ^ 27)
A 18/(b ^ 58)
B 45/(b ^ 58)
D * (45a ^ 16 * c ^ 34)/(b ^ 27)

Answer 1

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:** \(\frac{45a^{16} c^{34}}{b^{27}}\)