$$\begin{aligned} \text{if }6^m = 46656, \\\text{ What is the value of }6^{m-2} \end{aligned}$$

Answer: Option B

Explanation:

$$\begin{aligned} = (256)^{\frac{5}{4}} = (4^4)^{\frac{5}{4}} = 4^5 = 1024 \end{aligned}$$

Answer: Option B

Explanation:

$$\begin{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\ &\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\ &\Rightarrow x - 2 = -(x - 7)\\\\ &\Rightarrow x - 2 = -x + 7\\\\ &\Rightarrow x-2 = -x + 7\\\\ &\Rightarrow 2x = 9\\\\ &\Rightarrow x = \dfrac{9}{2} = 4.5 \end{align}$$

Answer: Option D

Explanation:

We know that $$\begin{aligned} (11)^2 = 121 \end{aligned}$$
So, putting values in said equation we get,
$$\begin{aligned} (11-1)^{2 + 1} = (10)^3 = 1000 \end{aligned}$$

Answer: Option D

Explanation:

$$\begin{aligned}3^{x-y} = 27 = 3^3 <=> x-y = 3 \text{... (i)}\\ 3^{x+y} = 243 = 3^5 <=> x+y = 5 \text{... (ii)} \\ \text{ adding (i) and (ii)} => 2x = 8 \\ => x = 4 \end{aligned}$$

Answer: Option C

Explanation:

$$\begin{aligned} 6^{m-2}\\ = \dfrac{6^m}{6^2}\\ = \dfrac{46656}{6^2}\\ = \dfrac{46656}{36} = 1296 \end{aligned}$$