If $2^{n+4} - 2^{n+2} = 3$, then $n$ is equal to

### Concept & Formula The problem tests your ability to factorize exponential expressions by isolating common terms using the **Laws of Indices**. The fundamental rule used for factoring here is: $$ a^{m+n} = a^m \times a^n $$ ### Step-by-Step Solution * **Step 1:** Break down the exponents to reveal a common factor. $2^{n+4}$ can be written as $2^{n+2} \times 2^2$. Substitute this into the original equation: $2^{n+2} \times 2^2 - 2^{n+2} = 3$ * **Step 2:** Factor out the common term, which is $2^{n+2}$. $2^{n+2} (2^2 - 1) = 3$ * **Step 3:** Simplify the expression inside the parentheses. $2^{n+2} (4 - 1) = 3$ $2^{n+2} (3) = 3$ * **Step 4:** Isolate the exponential term by dividing both sides by 3. $2^{n+2} = \frac{3}{3}$ $2^{n+2} = 1$ * **Step 5:** Express $1$ as a power of $2$ to equate the bases. Any non-zero number to the power of $0$ is $1$, so $1 = 2^0$. $2^{n+2} = 2^0$ * **Step 6:** Equate the exponents. $n + 2 = 0$ $n = -2$ ### Exam Strategy & Shortcut Instead of doing algebraic manipulation, you can use **Option Elimination**. Since the equation evaluates to a small integer ($3$), $n$ must be a negative number or zero to keep the powers of $2$ small. Try $n = -2$: $2^{-2+4} - 2^{-2+2} = 2^2 - 2^0 = 4 - 1 = 3$. This perfectly matches the right-hand side. The answer is found in just a few seconds. ### Common Pitfall A common mistake is incorrectly subtracting the bases or powers directly, mistakenly turning $2^{n+4} - 2^{n+2}$ into something invalid like $2^2$. Remember that you can only add or subtract coefficients of identical exponential terms. ### Final Answer **Therefore, the correct answer is -2.**