If $(10^{12} + 25)^2 - (10^{12} - 25)^2 = 10^n$, then the value of $n$ is

### Concept & Formula This problem is a direct application of standard **Algebraic Identities**. Instead of trying to calculate massively huge exponents, recognize the structure of the equation. The expression is in the form of: $$ (a + b)^2 - (a - b)^2 $$ The algebraic shortcut for this expansion is: $$ (a + b)^2 - (a - b)^2 = 4ab $$ ### Step-by-Step Solution * **Given:** * The equation is $(10^{12} + 25)^2 - (10^{12} - 25)^2 = 10^n$ * By matching the pattern, let $a = 10^{12}$ and $b = 25$. * **Calculation:** * Apply the identity $(a + b)^2 - (a - b)^2 = 4ab$: $$ = 4 \cdot (10^{12}) \cdot (25) $$ * Rearrange the multiplication to group the constants: $$ = (4 \cdot 25) \cdot 10^{12} $$ * Simplify the constants: $$ = 100 \cdot 10^{12} $$ * Convert $100$ into a power of $10$: $$ = 10^2 \cdot 10^{12} $$ * Apply the law of exponents ($x^m \cdot x^n = x^{m+n}$): $$ = 10^{2 + 12} = 10^{14} $$ * The problem states this equals $10^n$. Therefore: $$ 10^{14} = 10^n \implies n = 14 $$ ### Exam Strategy & Shortcut Memorize the twin identities: 1. $(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)$ 2. $(a+b)^2 - (a-b)^2 = 4ab$ When you see the minus sign between the squares, immediately multiply the two inner terms by $4$. $4 \times 25$ is $100$ (which is $10^2$). Adding this power of $2$ to the existing power of $12$ instantly gives you $14$. ### Common Pitfall Students who forget the identity often attempt to partially expand the squares and get lost in the zero-counting, leading to arithmetic mistakes. Another error is writing $100 \cdot 10^{12}$ as $100^{12}$ or misapplying exponent laws when multiplying bases. ### Final Answer **Therefore, the correct answer is 14.**