If \(\mathrm{8}^{\mathrm{2k}} = \left(\frac{1}{16}\right)^{\mathrm{k} - \frac{3}{4}}\), what is the value of k?

1. INFER the solution strategy

• Given: \(8^{2k} = (\frac{1}{16})^{k - \frac{3}{4}}\)
• Key insight: Since we have different bases (8 and 16), we need to express both sides using a common base
• Strategy: Use base 2 since both 8 and 16 are powers of 2

2. TRANSLATE both sides to the common base

• Express in terms of base 2:
  • \(8 = 2^3\)
  • \(16 = 2^4\), so \(\frac{1}{16} = 2^{-4}\)
• Rewrite the equation: \((2^3)^{2k} = (2^{-4})^{k - \frac{3}{4}}\)

3. SIMPLIFY using exponent rules

• Apply the power rule \((a^m)^n = a^{mn}\) to both sides:
  • Left side: \(2^{3 \times 2k} = 2^{6k}\)
  • Right side: \(2^{-4 \times (k - \frac{3}{4})} = 2^{-4k + 3}\)
• Now we have: \(2^{6k} = 2^{-4k + 3}\)

4. INFER that equal bases mean equal exponents

• Since both sides have base 2, the exponents must be equal:
  \(6k = -4k + 3\)

5. SIMPLIFY the linear equation

• Add 4k to both sides: \(6k + 4k = 3\)
• Combine like terms: \(10k = 3\)
• Divide by 10: \(k = \frac{3}{10}\)

Answer: B) 3/10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the need for a common base strategy and instead try to work with the original bases directly.

They might attempt something like taking logarithms of both sides without proper setup, or try to manipulate \(8^{2k}\) and \((\frac{1}{16})^{k - \frac{3}{4}}\) as separate entities. This leads to confusion and they often abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the common base strategy but make arithmetic errors in the exponent manipulations.

Common mistakes include:

• Incorrectly expanding \((-4) \times (k - \frac{3}{4})\) as \(-4k - 3\) instead of \(-4k + 3\)
• Sign errors when moving terms in the linear equation
• Arithmetic errors when solving \(10k = 3\)

These calculation errors typically lead them to select Choice A (1/5) or Choice D (1/2).

The Bottom Line:

This problem tests whether students can recognize that exponential equations with different bases require a common base strategy, then execute the multi-step algebraic manipulation accurately. The conceptual insight about common bases is crucial - without it, students get stuck immediately.