The given exponential function is:

$f(t) = 41000 \left( \frac{1}{2} \right)^{\frac{t}{8}}$

We need to rewrite it in the form:

$f(t) = a \cdot b^{\frac{t}{15}}$

Step-by-step process:

1. Compare the given form with the desired form: The desired form has an exponent of $\frac{t}{15}$, but the original function has $\frac{t}{8}$ as the exponent base.

2. Rewrite the base: To match the two forms, we need to manipulate the exponent. The current exponent is $\frac{t}{8}$, so we need to adjust this to be $\frac{t}{15}$.

   Start with the base of the exponential function: $\left( \frac{1}{2} \right)^{\frac{t}{8}} = \left( \left( \frac{1}{2} \right)^{\frac{1}{8}} \right)^t$

3. Adjust the base to match the desired exponent: We need to express this as $b^{\frac{t}{15}}$. To do so, we need: $\left( \frac{1}{2} \right)^{\frac{1}{8}} = b^{\frac{1}{15}}$ To find $b$, we raise both sides to the power of 15: $b = \left( \frac{1}{2} \right)^{\frac{15}{8}}$

4. Calculate $b$: Now, calculate $\left( \frac{1}{2} \right)^{\frac{15}{8}}$: $b = \left( \frac{1}{2} \right)^{1.875} = 0.1240 \quad (\text{rounded to four decimal places})$

5. Adjust $a$: Now that we have the new base $b$, we need to adjust the constant coefficient $a$. To do this, we recognize that:

   $f(t) = 41000 \cdot \left( \frac{1}{2} \right)^{\frac{t}{8}} = a \cdot b^{\frac{t}{15}}$

   So, $a$ is: $a = 41000 \cdot b^{-\frac{t}{15}} = \frac{41000}{b^{\frac{t}{15}}}$

   Since $b^{\frac{t}{15}} = \left( \frac{1}{2} \right)^{\frac{t}{15}}$, we use the relation and calculate it:

   $a = 41000 \cdot (0.1240)^{-1} = 41000 \cdot 8.0645 \approx 330645.8000$

Final Answer:

The function in the desired form is:

$f(t) = 330645.8000 \cdot 0.1240^{\frac{t}{15}}$

Would you like a more detailed breakdown or help with another question? Here's a tip: Exponential functions are powerful for modeling growth and decay, and converting between different forms can be useful for easier interpretation of data.

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