3 which expression is equivalent to $4^{-9}$?\na $4^{9}\\cdot4^{-1}$\nb $(4^{-9})^{0}$\nc $\\frac{1}{4^{9}}$\…

Explanation:

Step1: Recall negative exponent rule

For any non-zero base $a$ and integer $n$, $a^{-n}=\frac{1}{a^n}$. So $4^{-9}=\frac{1}{4^9}$.

Step2: Evaluate Option A

Use product rule: $a^m \cdot a^n=a^{m+n}$. $4^9 \cdot 4^{-1}=4^{9+(-1)}=4^8$, not equal to $4^{-9}$.

Step3: Evaluate Option B

Use zero exponent rule: $(a^m)^0=1$ (for $a\neq0$). $(4^{-9})^0=1$, not equal to $4^{-9}$.

Step4: Evaluate Option C

Match with negative exponent result: $\frac{1}{4^9}=4^{-9}$, which is equivalent.

Step5: Evaluate Option D

Simplify fraction with exponents: $\frac{4^9}{4^0}=4^{9-0}=4^9$, not equal to $4^{-9}$.

Answer:

C. $\frac{1}{4^9}$