$25^{2.7} \times 5^{4.2} \div 5^{5.4} = 25^{x}$

### Concept & Formula The problem tests the fundamental **Laws of Indices**, specifically converting different bases to a common prime base to equate exponents. The key formulas used are: $$ (a^m)^n = a^{m \times n} $$ $$ a^m \times a^n = a^{m+n} $$ $$ a^m \div a^n = a^{m-n} $$ ### Step-by-Step Solution * **Step 1:** Convert all bases to the common prime base of 5. $25^{2.7} = (5^2)^{2.7} = 5^{5.4}$ The Right Hand Side (RHS) is $25^x = (5^2)^x = 5^{2x}$ * **Step 2:** Rewrite the full equation using the common base. $5^{5.4} \times 5^{4.2} \div 5^{5.4} = 5^{2x}$ * **Step 3:** Apply the laws of multiplication and division of indices. $5^{5.4 + 4.2 - 5.4} = 5^{2x}$ $5^{4.2} = 5^{2x}$ * **Step 4:** Equate the exponents since the bases are equal. $2x = 4.2$ $x = 2.1$ ### Exam Strategy & Shortcut Notice that $25^{2.7}$ is exactly $5^{5.4}$. When you see $5^{5.4} \times \dots \div 5^{5.4}$, the first and last terms on the Left Hand Side (LHS) immediately cancel each other out (i.e., $a \times b \div a = b$). This instantly simplifies the LHS to $5^{4.2}$. Equating this to $25^x$ (which is $5^{2x}$) directly gives $2x = 4.2$, or $x = 2.1$. This mental shortcut saves crucial seconds. ### Common Pitfall A frequent error is forgetting to convert the target base (25) to the common base (5). Students might calculate the LHS exponent as 4.2 and incorrectly assume $x = 4.2$, failing to account for the $25^x$ turning into $5^{2x}$. ### Final Answer **Therefore, the correct answer is None of these.**