can you use an approach closer to grade 10 and 11 exponential function stuff/ keep it simple.

Sure, let's simplify the explanation and approach for grade 10 and 11 levels.

### Question 9: Simplify the expression \(( - 3a^{-2}b^2)^{-3} \cdot ( - a^{-3}b^4)\)

1. Simplify each term separately:
   \[
   ( - 3a^{-2}b^2)^{-3} = (-3)^{-3} \cdot (a^{-2})^{-3} \cdot (b^2)^{-3}
   \]
   \[
   = -\frac{1}{27} \cdot a^{6} \cdot b^{-6}
   \]
   \[
   = -\frac{a^6}{27b^6}
   \]

   For the second term:
   \[
   ( - a^{-3}b^4) = -a^{-3}b^4
   \]

2. Combine the two terms:
   \[
   -\frac{a^6}{27b^6} \cdot (-a^{-3}b^4) = \frac{a^6 \cdot a^{-3}}{27b^6} \cdot b^4
   \]
   \[
   = \frac{a^{6-3} \cdot b^{4-6}}{27} = \frac{a^3 b^{-2}}{27}
   \]
   \[
   = \frac{a^3}{27b^2}
   \]

Therefore, the simplified expression is:
\[
\boxed{\frac{a^3}{27b^2}}
\]

### Question 10: Analyze the function \( f(x) = -3\left( \frac{1}{2} \right)^x + 2 \)

#### a) Domain and Range:

- **Domain**: The domain of \( f(x) \) is all real numbers \(\mathbb{R}\), because exponential functions are defined for all real numbers.

- **Range**: The range is determined by analyzing the behavior of the function. Since \(\left( \frac{1}{2} \right)^x\) is always positive and tends to 0 as \( x \to \infty \) and grows very large as \( x \to -\infty \), the function \( -3\left( \frac{1}{2} \right)^x \) will range from \(-3\) to \(-\infty\). Adding 2 to this will range from \(-1\) to \(2\). Therefore, the range is:
  \[
  (-1, 2]
  \]

#### b) y-intercept:

The y-intercept occurs when \( x = 0 \):
\[
f(0) = -3\left( \frac{1}{2} \right)^0 + 2 = -3(1) + 2 = -1
\]

Therefore, the y-intercept is \((0, -1)\).

#### c) Whether the function is increasing or decreasing:

- **Increasing/Decreasing**: The base \(\frac{1}{2}\) is less than 1, indicating the exponential function \(\left( \frac{1}{2} \right)^x\) is decreasing. Multiplying by \(-3\) and adding 2 maintains a decreasing behavior.
- **Rate**: The rate at which the function decreases depends on the exponential rate of \(\left( \frac{1}{2} \right)^x\), which decreases exponentially.

Thus, the function is decreasing at an exponentially decreasing rate.

### Question 11: Determine the equation of the exponential function with base 4

Given:
- Reflected in the x-axis
- Asymptote at y = 3
- Passes through point (1, -5)

Let's start by writing the general form of an exponential function with base 4:
\[
f(x) = a \cdot 4^x + b
\]

Reflecting in the x-axis:
\[
f(x) = -a \cdot 4^x + b
\]

Since the asymptote is y = 3, \( b = 3 \):
\[
f(x) = -a \cdot 4^x + 3
\]

It passes through (1, -5), so:
\[
-5 = -a \cdot 4^1 + 3
\]
\[
-5 = -4a + 3
\]
\[
-8 = -4a
\]
\[
a = 2
\]

Therefore, the function is:
\[
f(x) = -2 \cdot 4^x + 3
\]

If you need further details or have any questions, feel free to ask! Here are 5 questions you might be interested in:

1. How do you solve logarithmic equations?
2. What is the derivative of an exponential function?
3. How do you find the inverse of a function?
4. What are the applications of exponential functions in real life?
5. How can you determine the concavity of a function?

**Tip**: When dealing with exponential functions, always pay attention to the base and transformations like reflections and translations to determine the function's behavior.

Explore simplified explanations and analyses of exponential functions suitable for Grades 10 and 11 students. Learn to simplify complex exponential expressions and analyze function behaviors, including domain, range, and growth rates.

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