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Answer this test using exponential function 

Let's break down and solve the provided problems step-by-step.

### 1) \( f(x) = \frac{1}{3} \cdot 6^x \) at \( x = 2 \)

\[
f(2) = \frac{1}{3} \cdot 6^2 = \frac{1}{3} \cdot 36 = 12
\]

### 2) \( f(n) = 10 \cdot 2^n \) at \( n = 5 \)

\[
f(5) = 10 \cdot 2^5 = 10 \cdot 32 = 320
\]

### 3) \( f(n) = 10 \cdot 2^n \) at \( n = -2 \)

\[
f(-2) = 10 \cdot 2^{-2} = 10 \cdot \frac{1}{4} = 2.5
\]

### 4) \( g(x) = \frac{1}{5} \left( \frac{1}{3} \right)^x \) at \( x = 3 \)

\[
g(3) = \frac{1}{5} \left( \frac{1}{3} \right)^3 = \frac{1}{5} \cdot \frac{1}{27} = \frac{1}{135}
\]

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### Solve each equation

### 5) \( 4^{3x - 2} = 1 \)

Since \( 4^0 = 1 \), set the exponent equal to 0:

\[
3x - 2 = 0 \implies 3x = 2 \implies x = \frac{2}{3}
\]

### 6) \( 4^{2p} = 4^{-2p - 1} \)

Since the bases are the same, set the exponents equal:

\[
2p = -2p - 1 \implies 4p = -1 \implies p = -\frac{1}{4}
\]

### 7) \( 6^{-2a} = 6^{2 - 3a} \)

Equating the exponents:

\[
-2a = 2 - 3a \implies a = 2
\]

### 8) \( 2^{2x + 2} = 2^{3x} \)

Equating the exponents:

\[
2x + 2 = 3x \implies x = 2
\]

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Let me know if you'd like more details on any specific step! 

Here are 5 follow-up questions for deeper exploration:
1. How do you simplify expressions with fractional exponents?
2. What happens to an equation when the base is negative?
3. Can you solve similar problems using logarithms?
4. What are the properties of exponential functions in graphing?
5. How can you apply exponential functions in real-world problems?

**Tip**: Always check the bases when working with exponential equations—matching them simplifies solving significantly!

Evaluate each function and solve the exponential equations provided.

This page provides a step-by-step solution to exponential functions and equations, including key formulas such as f(x) = 1/3 * 6^x and g(x) = 1/5 * (1/3)^x, ideal for high school students in Grades 10-12.

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