Question

#7 extra - spicy! which expression is equivalent to $\frac{1}{4}(4a^{2}b)^{3}(3a^{4}b^{4})$ for all values of $a$ and $b$ where the expression is defined?
a. $48a^{6}b^{12}$
b. $12a^{8}b^{7}$
c. $48a^{10}b^{7}$
d. $12a^{10}b^{7}$

Explanation:

Step1: Expand the power term

First, apply the power rule \((x^m y^n)^p = x^{m \cdot p} y^{n \cdot p}\) to \((4a^2b)^3\):
$$(4a^2b)^3 = 4^3 \cdot (a^2)^3 \cdot b^3 = 64a^6b^3$$

Step2: Multiply by the remaining term

Multiply the result by \(3a^4b^4\):
$$64a^6b^3 \cdot 3a^4b^4 = (64 \cdot 3) \cdot a^{6+4} \cdot b^{3+4} = 192a^{10}b^7$$

Step3: Apply the leading fraction

Multiply by \(\frac{1}{4}\):
$$\frac{1}{4} \cdot 192a^{10}b^7 = \frac{192}{4}a^{10}b^7 = 48a^{10}b^7$$

Answer:

c. \(48a^{10}b^7\)