If (3^33 + 3^33 + 3^33) × (2^33 + 2^33) = 6^x, then what is the value of x?

Introduction / Context:
Questions involving powers of integers are common in quantitative aptitude. They often test your understanding of how exponents behave under addition and multiplication. In this problem, you are given an expression built from powers of 2 and 3, and you must express the entire product as a single power of 6, namely 6^x. This requires recognising repeated terms, factoring them, and then applying the laws of indices systematically.

Given Data / Assumptions:

• The given product is (3^33 + 3^33 + 3^33) × (2^33 + 2^33).
• This product is equal to 6^x for some exponent x.
• We must find the value of x based on the rules of exponents.
• All exponents are real numbers, and standard index laws apply.

Concept / Approach:
The key step is to notice that 3^33 appears three times in the first bracket, and 2^33 appears twice in the second bracket. Thus the expression inside each bracket can be factored by counting how many identical terms there are. Specifically, three copies of 3^33 give 3 × 3^33, and two copies of 2^33 give 2 × 2^33. Using a^m × a^n = a^(m+n), we can then combine the powers of 2 and 3 into a single power of 6, because 6 = 2 × 3.

Step-by-Step Solution:
Step 1: Simplify the first bracket: 3^33 + 3^33 + 3^33 = 3 × 3^33. Step 2: Use the law of exponents: 3 × 3^33 = 3^1 × 3^33 = 3^(1 + 33) = 3^34. Step 3: Simplify the second bracket: 2^33 + 2^33 = 2 × 2^33. Step 4: Apply the same law: 2 × 2^33 = 2^1 × 2^33 = 2^(1 + 33) = 2^34. Step 5: Multiply the simplified brackets: 3^34 × 2^34 = (3 × 2)^34 = 6^34. Step 6: Compare with the given form 6^x. Since the expression equals 6^34, we have x = 34.

Verification / Alternative check:
You can check the logic by considering a smaller exponent. For instance, if the pattern were (3^2 + 3^2 + 3^2)(2^2 + 2^2), you would have 3 × 3^2 = 3^3 and 2 × 2^2 = 2^3, and the product would be 6^3. This confirms that counting the repeated terms and then combining exponents is a sound approach. The same reasoning applies to an exponent as large as 33, giving x = 34.

Why Other Options Are Wrong:

• Option b (35) incorrectly suggests adding an extra one to the exponent, perhaps from miscounting terms.
• Option c (33) uses the original exponent without considering that the repeated terms add one extra power of each base.
• Option d (33.5) introduces a fractional exponent, which is not supported by the integer counting of factors in the problem.
• Option e (32) understates the exponent and could come from subtracting instead of adding when combining powers.

Common Pitfalls:
A typical error is to treat 3^33 + 3^33 as 3^66, which is incorrect because addition of powers does not follow the same law as multiplication. Another mistake is to forget that multiplying by 3 or 2 adds one to the exponent rather than changing the base. To avoid these problems, remember that a^m + a^m is addition, not multiplication, and can be factored as a^m(1 + 1). Then apply exponent rules only to products, not to sums.

Final Answer:
The exponent of 6 in the expression is 34.