Simplify $\sqrt{\frac{8^2 \times 4^{n + 1}}{2^{2n} \times 16}}$

Analyze the question: We need to simplify a surd expression with powers of 2. The trick is to write every number as a power of 2.

Concept: Use the laws of indices:

• (aᵐ)ⁿ = aᵐⁿ
• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• √(aᵐ) = a^(m/2)

Solution:

Rewrite each term in base 2:

• 8² = (2³)² = 2⁶
• 4ⁿ⁺¹ = (2²)ⁿ⁺¹ = 2²ⁿ⁺²
• 2²ⁿ stays as 2²ⁿ
• 16 = 2⁴

Numerator: 2⁶ × 2²ⁿ⁺² = 2²ⁿ⁺⁸

Denominator: 2²ⁿ × 2⁴ = 2²ⁿ⁺⁴

Divide: 2²ⁿ⁺⁸ ÷ 2²ⁿ⁺⁴ = 2⁴ = 16

Now take the square root: √16 = 4

Answer: The expression simplifies to 4, which is option C.

Tip: When you see a mix of 2, 4, 8, 16, convert everything to base 2 first. The "n" terms cancel out, which is a sign you are on the right track. Also, don't forget the final square root. Stopping at 16 (option A) is the common trap!