The exponential function h is defined by \(\mathrm{h(x) = 12 \cdot b^x}\), where b is a positive constant. If \(\mathrm{h(4)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(x) = 12 \cdot b^x}\) (exponential function)
  • \(\mathrm{h(4) = 3{,}072}\)
  • Need to find \(\mathrm{h(3)}\)

2. INFER the solution strategy

• Key insight: Since we know \(\mathrm{h(4)}\), we can work backwards to find the base \(\mathrm{b}\)
• Once we have \(\mathrm{b}\), we can calculate \(\mathrm{h(3)}\) directly
• This is more efficient than trying to find a relationship between \(\mathrm{h(3)}\) and \(\mathrm{h(4)}\) directly

3. SIMPLIFY to find the base b

• Start with \(\mathrm{h(4) = 12 \cdot b^4 = 3{,}072}\)
• Divide both sides by 12: \(\mathrm{b^4 = 3{,}072 \div 12 = 256}\)
• Take the fourth root: \(\mathrm{b = 256^{1/4} = 4}\)
  (We can verify: \(\mathrm{4^4 = 4 \times 4 \times 4 \times 4 = 256}\) ✓)

4. SIMPLIFY to calculate h(3)

• Now substitute \(\mathrm{b = 4}\) into \(\mathrm{h(3)}\):
• \(\mathrm{h(3) = 12 \cdot b^3 = 12 \cdot 4^3}\)
• Calculate: \(\mathrm{4^3 = 4 \times 4 \times 4 = 64}\)
• Final calculation: \(\mathrm{h(3) = 12 \times 64 = 768}\)

Answer: C) 768

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the base \(\mathrm{b}\) first. Instead, they try to establish a direct relationship between \(\mathrm{h(3)}\) and \(\mathrm{h(4)}\) without finding \(\mathrm{b}\), or they attempt to guess-and-check with the answer choices. This leads to confusion and abandoning systematic solution, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{3{,}072 \div 12}\), incorrectly computing it as something other than 256, or they struggle with finding the fourth root of 256. These calculation errors propagate through the rest of the problem. This may lead them to select Choice A (192) or Choice B (384) depending on the specific arithmetic mistake.

The Bottom Line:

This problem requires students to understand that exponential functions work both ways - you can use a known output to find the base, then use that base to find other outputs. The key strategic insight is recognizing this backwards-then-forwards approach rather than trying to find shortcuts.