$$ 49 \times 49 \times 49 \times 49 = 7^x $$

Concept & Formula This question evaluates the ability to express composite numbers using prime bases and applying the exponent rules for multiplication. $$ (a^m)^n = a^{m \times n} $$ $$ a^m \times a^n = a^{m+n} $$ Step-by-Step Solution * The right side of the equation uses a base of 7. To equate both sides, convert the numbers on the left side to base 7. * We know that $$ 49 = 7^2 $$. * Rewrite the original expression by replacing every 49 with $$ 7^2 $$: $$ 7^2 \times 7^2 \times 7^2 \times 7^2 = 7^x $$ * Apply the product rule for exponents. Since the bases are identical, add all the exponents together: $$ 2 + 2 + 2 + 2 = 8 $$ * This simplifies the left side to $$ 7^8 $$. * Equate both sides: $$ 7^8 = 7^x $$ * Therefore, equating the exponents gives $$ x = 8 $$. Exam Strategy & Shortcut Recognize the relationship between 49 and 7 immediately. Since there are four instances of 49 being multiplied, and each 49 is a "square" ($$7^2$$), simply multiply the number of terms (4) by the power (2). $$4 \times 2 = 8$$. You arrive at the correct exponent without writing down the intermediate steps. Common Pitfall A frequent mistake is to just count the number of times 49 appears (which is 4) and blindly select option (a). Remember that the base changes from 49 to 7, so you must account for the square factor. $$49^4$$ is not equal to $$7^4$$. Final Answer **Therefore, the correct answer is 8.**