https://r2.erweima.ai/mathbot/image/17254885715588h17nbw4.jpg

Let's analyze and simplify the expression for \( A \) given in the image:

\[
A = \frac{2^{x+4} + 2 \cdot 2^x}{3 \cdot 2^{x+3}}
\]

### Step 1: Simplify the Numerator
The numerator is \( 2^{x+4} + 2 \cdot 2^x \).

We can rewrite the second term using the properties of exponents:

\[
2 \cdot 2^x = 2^1 \cdot 2^x = 2^{x+1}
\]

So the numerator becomes:

\[
2^{x+4} + 2^{x+1}
\]

### Step 2: Factor the Numerator
Now, factor out the common factor \( 2^x \) from both terms:

\[
2^x(2^4 + 2^1) = 2^x(16 + 2) = 2^x \cdot 18
\]

### Step 3: Simplify the Denominator
The denominator is \( 3 \cdot 2^{x+3} \).

### Step 4: Simplify the Fraction
Now, simplify the expression by canceling out \( 2^x \) from both the numerator and the denominator:

\[
A = \frac{2^x \cdot 18}{3 \cdot 2^{x+3}}
\]

This simplifies further because:

\[
A = \frac{18}{3 \cdot 2^3} = \frac{18}{3 \cdot 8} = \frac{18}{24} = \frac{3}{4}
\]

### Final Answer:
\[
A = \frac{3}{4}
\]

Would you like more details or have any questions?

### Related Questions:
1. How can you apply the properties of exponents to simplify other expressions?
2. What are some other examples where factoring can simplify an expression?
3. How can the simplification process change if the powers were different in the original problem?
4. What are some strategies for simplifying complex algebraic fractions?
5. How does the method of canceling out common factors work in general cases?

### Tip:
Always look for common factors in the numerator and denominator to simplify expressions efficiently.

Simplify the expression A = (2^(x+4) + 2(2^x)) / (3(2^(x+3))).

Learn how to simplify the exponential expression A = (2^(x+4) + 2(2^x)) / (3(2^(x+3))) using exponent rules and factoring. This step-by-step solution is ideal for students in Grades 9-12.

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