$(0.04)^{2} \div (0.008) \times (0.2)^{6} = (0.2)^{x}$

### Concept & Formula The problem tests your ability to recognize decimal powers and apply the **Laws of Indices**. It requires converting varying decimal bases into a uniform decimal base. The governing rules are: $$ (a^m)^n = a^{m \times n} $$ $$ a^m \div a^n \times a^p = a^{m-n+p} $$ ### Step-by-Step Solution * **Step 1:** Observe that the target base on the Right Hand Side is 0.2. Express the other decimals as powers of 0.2. $0.04 = (0.2)^2$ $0.008 = (0.2)^3$ * **Step 2:** Substitute these into the original terms. $(0.04)^2 = ((0.2)^2)^2 = (0.2)^4$ The equation becomes: $(0.2)^4 \div (0.2)^3 \times (0.2)^6 = (0.2)^x$ * **Step 3:** Apply the laws of indices from left to right. $(0.2)^{4 - 3 + 6} = (0.2)^x$ $(0.2)^{1 + 6} = (0.2)^x$ $(0.2)^7 = (0.2)^x$ * **Step 4:** Equate the exponents. $x = 7$ ### Exam Strategy & Shortcut Instead of working with decimals which can be prone to error, you can focus strictly on the significant digit (2) and its place value. Recognize $0.04$ as $2^2$ shifted two decimal places, which aligns with $(0.2)^2$. Immediately write the exponent arithmetic: $2 \times 2 - 3 + 6 = 7$. By skipping the formal notation and mapping decimals to powers of $0.2$, you arrive at $x = 7$ in just seconds. ### Common Pitfall A major pitfall is applying the multiplication before the division, effectively calculating $(0.2)^4 \div ((0.2)^3 \times (0.2)^6) = (0.2)^{4 - 9} = (0.2)^{-5}$. Always adhere strictly to left-to-right processing for operators of the same precedence (Division and Multiplication). ### Final Answer **Therefore, the correct answer is None of these.**