Question

1. simplify the expression. ((4x^{3})^{-2})

Explanation:

Step1: Apply power of a product rule

$(ab)^n = a^n b^n$, so $(4x^3)^{-2} = 4^{-2} \cdot (x^3)^{-2}$

Step2: Apply negative exponent rule

$a^{-n} = \frac{1}{a^n}$, so $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

Step3: Apply power of a power rule

$(a^m)^n = a^{m \cdot n}$, so $(x^3)^{-2} = x^{3 \cdot (-2)} = x^{-6} = \frac{1}{x^6}$

Step4: Multiply the results

Multiply the two simplified terms: $\frac{1}{16} \cdot \frac{1}{x^6}$

Answer:

$\frac{1}{16x^6}$