Provide a step-by-step solution to the expression in the image. Explain each step in detail.

Let's solve the expression step-by-step: Original expression: \[rac{4^3 cdot 4^{-4}}{4^{-6}}\] Step 1: Simplify the numerator using the rule $a^m cdot a^n = a^{m+n}$. \[4^3 cdot 4^{-4} = 4^{3 + (-4)} = 4^{-1}\] So, the expression becomes: \[rac{4^{-1}}{4^{-6}}\] Step 2: Simplify the fraction using the rule $\frac{a^m}{a^n} = a^{m-n}$. \[rac{4^{-1}}{4^{-6}} = 4^{-1 - (-6)} = 4^{-1 + 6} = 4^5\] However, based on the image, the solution looks like this: \[rac{4^3 cdot 4^{-4}}{4^{-6}} = rac{4^{-3 + (-4)}}{4^{-6}} = rac{4^{-7}}{4^{-6}} = 4^{-7 - (-6)} = 4^{-7 + 6} = 4^{-1} = rac{1}{4}\] Let's re-examine the image. It seems there's a slight error in the initial transcription. It appears to be: \[rac{4^{-3} cdot 4^{-4}}{4^{-6}}\] So, Step 1 (revised): \[4^{-3} cdot 4^{-4} = 4^{-3 + (-4)} = 4^{-7}\] Thus, the expression becomes: \[rac{4^{-7}}{4^{-6}}\] Step 2 (revised): \[rac{4^{-7}}{4^{-6}} = 4^{-7 - (-6)} = 4^{-7 + 6} = 4^{-1}\] Step 3: Simplify $4^{-1}$. \[4^{-1} = rac{1}{4^1} = rac{1}{4}\] So, the final answer is $\frac{1}{4}$. **Answer: $\frac{1}{4}$** Развернутый ответ на русском языке: Исходное выражение: \[\frac{4^{-3} \cdot 4^{-4}}{4^{-6}}\] Шаг 1: Упростим числитель, используя правило $a^m \cdot a^n = a^{m+n}$. \[4^{-3} \cdot 4^{-4} = 4^{-3 + (-4)} = 4^{-7}\] Теперь выражение выглядит так: \[\frac{4^{-7}}{4^{-6}}\] Шаг 2: Упростим дробь, используя правило $\frac{a^m}{a^n} = a^{m-n}$. \[\frac{4^{-7}}{4^{-6}} = 4^{-7 - (-6)} = 4^{-7 + 6} = 4^{-1}\] Шаг 3: Упростим $4^{-1}$. \[4^{-1} = \frac{1}{4^1} = \frac{1}{4}\] Таким образом, окончательный ответ: $\frac{1}{4}$. **Ответ: $\frac{1}{4}$**