If 3^10 × 27^2 = 9^2 × 3^n, express both sides as powers of 3 and find the value of the exponent n.

Introduction / Context:
This problem tests exponent rules and the ability to express different bases as powers of a common base. In this case all numbers involved, 3, 27 and 9, can be written as powers of 3. Once everything is converted to 3 raised to some exponent, comparing exponents on both sides of the equation directly yields the value of n. Questions like this are standard in quantitative aptitude and algebra topics on indices and powers.

Given Data / Assumptions:

• Equation: 3^10 × 27^2 = 9^2 × 3^n.

• All quantities are positive real numbers.

• We need to determine n so that the equality holds.

Concept / Approach:
The main ideas are: write 27 and 9 as powers of 3, then apply the law a^m × a^n = a^(m+n). On the left side, we express 27^2 in terms of 3. On the right side, we express 9^2 in terms of 3. With everything written as 3 raised to some exponent, equality of powers implies equality of exponents because the base is the same and non zero. Solving the resulting simple linear equation gives n.

Step-by-Step Solution:
Step 1: Note that 27 = 3^3. Therefore 27^2 = (3^3)^2 = 3^(3×2) = 3^6. Step 2: Compute the exponent on the left: 3^10 × 3^6 = 3^(10 + 6) = 3^16. Step 3: Note that 9 = 3^2. Therefore 9^2 = (3^2)^2 = 3^(2×2) = 3^4. Step 4: On the right side, we have 9^2 × 3^n = 3^4 × 3^n = 3^(4 + n). Step 5: Since 3^16 = 3^(4 + n) and the base 3 is the same and positive, the exponents must be equal: 16 = 4 + n. Step 6: Solve for n: n = 16 − 4 = 12.

Verification / Alternative check:
As a quick check, substitute n = 12 back into the equation. The right side becomes 9^2 × 3^12 = 3^4 × 3^12 = 3^16. The left side we already simplified to 3^16, so both sides match. No other value of n can satisfy the equality because exponents of the same positive base are unique. Therefore n = 12 is confirmed as correct.

Why Other Options Are Wrong:
If n were 10, the right side exponent would be 4 + 10 = 14, giving 3^14, which is smaller than 3^16. For n = 15, we would get 3^19, which is larger than 3^16. Values 20 and 8 similarly yield exponents 24 and 12, neither equal to 16. Only n = 12 makes the exponents match exactly and thus satisfies the equation.

Common Pitfalls:
Common mistakes include adding exponents incorrectly, for example writing 3^10 × 3^6 as 3^60, or miswriting 27 as 3^2 instead of 3^3. Another error is failing to convert 9 and 27 into powers of 3 and attempting to compare products numerically, which is time consuming and prone to error. Remembering the key identities and always expressing terms with a common base makes exponent comparison problems straightforward.

Final Answer:
The value of the exponent n that satisfies the equation is 12.