Write the function f, of, t, equals, 6700, left parenthesis, 0, point, 9, 6, right parenthesis, to the power t f(t)=6700(0.96)
t
in the form f, of, t, equals, a, b, start superscript, start fraction, t, divided by, 25, end fraction, end superscriptf(t)=ab
25
t
​

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To write the function \( f(t) = 6700(0.96)^t \) in the form \( f(t) = ab^{\frac{t}{25}} \), we need to express \( 0.96^t \) in the form \( b^{\frac{t}{25}} \).

### Step 1: Identify the equivalent base
We start by comparing the exponents in the two forms:

\[
0.96^t = \left(b^{\frac{1}{25}}\right)^t
\]

Taking the \( t \)-th root on both sides, we get:

\[
0.96 = b^{\frac{1}{25}}
\]

### Step 2: Solve for \( b \)
To solve for \( b \), raise both sides to the power of 25:

\[
b = 0.96^{25}
\]

### Step 3: Substitute into the original form
Now, substitute the value of \( b \) back into the equation:

\[
f(t) = 6700 \cdot \left(0.96^{25}\right)^{\frac{t}{25}}
\]

Thus, the function \( f(t) = 6700(0.96)^t \) can be written in the form \( f(t) = ab^{\frac{t}{25}} \) as:

\[
\boxed{f(t) = 6700 \cdot (0.96^{25})^{\frac{t}{25}}}
\]

Where \( a = 6700 \) and \( b = 0.96^{25} \).

### Final Answer:
\[
f(t) = 6700 \cdot (0.96^{25})^{\frac{t}{25}}
\]

Would you like more details on any part of the solution or have any questions?

### Questions:
1. How do you calculate \( 0.96^{25} \)?
2. What is the interpretation of the exponential form \( f(t) = ab^{\frac{t}{25}} \)?
3. How do you convert different exponential forms?
4. What is the significance of the base \( 0.96 \) in the context of decay?
5. How does the value of \( t \) affect the function \( f(t) \)?
6. Can this approach be generalized for any similar functions?
7. What are common applications of exponential decay in real life?
8. What if \( t \) were negative—how would it affect \( f(t) \)?

### Tip:
When dealing with exponential functions, it's useful to understand the properties of exponents, particularly when changing bases or rewriting forms for simplification.

Learn how to convert the exponential function f(t) = 6700(0.96)^t to the form f(t) = ab^(t/25) with a detailed step-by-step solution. This problem involves exponential functions and their applications, including exponential decay and the change of base formula. Suitable for high school students studying advanced algebra and calculus concepts.

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