The function g is defined by the equation \(\mathrm{g(t) = (1.09)^t}\). The equation can be rewritten as \(\mathrm{g(t) = b^{t/3}}\),...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(t) = (1.09)^t}\) (original form)
  • \(\mathrm{g(t) = b^{t/3}}\) (rewritten form)
  • Need to find the value of b

• What this tells us: Since both expressions equal g(t), they must be equal to each other.

2. INFER the solution strategy

• Since both expressions represent the same function g(t), we can set them equal:
  \(\mathrm{(1.09)^t = b^{t/3}}\)

• Key insight: We need to get both sides in a form where we can compare the bases directly.

3. SIMPLIFY using exponent rules

• Rewrite the right side using the rule \(\mathrm{a^{mn} = (a^m)^n}\):
  \(\mathrm{b^{t/3} = (b^{1/3})^t}\)

• Now our equation becomes:
  \(\mathrm{(1.09)^t = (b^{1/3})^t}\)

4. INFER the relationship between bases

• Since both sides have the same exponent t, and the expressions are equal for all values of t, the bases must be equal:
  \(\mathrm{1.09 = b^{1/3}}\)

5. SIMPLIFY to solve for b

• To isolate b, cube both sides:
  \(\mathrm{b = (1.09)^3}\)

6. SIMPLIFY the calculation

• Calculate \(\mathrm{(1.09)^2}\) first:
  \(\mathrm{(1.09)^2 = (1 + 0.09)^2 = 1 + 0.18 + 0.0081 = 1.1881}\)

• Then multiply by 1.09 again:
  \(\mathrm{b = 1.1881 \times 1.09 \approx 1.295}\)

• Comparing to answer choices: 1.295 is closest to 1.30

Answer: C (1.30)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that two expressions equal to the same function must be equal to each other, or they don't see the strategy of rewriting \(\mathrm{b^{t/3}}\) to match the exponential form.

Instead, they might try to substitute specific values of t or attempt algebraic manipulation that doesn't lead anywhere productive. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors when calculating \(\mathrm{(1.09)^3}\), particularly in the multi-step arithmetic.

Common calculation mistakes include errors in \(\mathrm{(1.09)^2}\) or in the final multiplication step. This may lead them to select Choice B (1.27) or Choice D (1.33) based on incorrect calculations.

The Bottom Line:

This problem tests whether students can recognize equivalent exponential expressions and systematically manipulate exponents to isolate an unknown base. The key insight is seeing that equal functions with the same variable must have equal expressions, then using exponent rules strategically to solve for the unknown.