6(12)^(x)+4=24(6^(x))+2^(x), maka nilai dari (x_(1)+x_(2))/(x_(1).x_(2))

To solve the equation \( 6(12)^x + 4 = 24(6^x) + 2^x \) and find the value of \(\frac{x_1 + x_2}{x_1 \cdot x_2}\), we'll follow these steps:

1. Simplify the equation.
2. Solve for \( x \).
3. Calculate the required expression.

### Step 1: Simplify the Equation

The given equation is:

\[
6(12)^x + 4 = 24(6^x) + 2^x
\]

We can rewrite \(12^x\) as \((2^2 \cdot 3)^x = 2^{2x} \cdot 3^x\) and \(6^x\) as \((2 \cdot 3)^x = 2^x \cdot 3^x\).

Thus, the equation becomes:

\[
6 \cdot (2^{2x} \cdot 3^x) + 4 = 24 \cdot (2^x \cdot 3^x) + 2^x
\]

### Step 2: Solve for \( x \)

Expanding and simplifying both sides, we find the values of \( x \).

### Step 3: Calculate \(\frac{x_1 + x_2}{x_1 \cdot x_2}\)

Let's proceed to find \( x \).The equation has multiple bases for the exponential terms (2, 3, and 6), making it challenging to solve directly. To approach this problem, let's use a substitution method. Let:

\[
y = 2^x
\]

Thus, \( 3^x = \left(\frac{3}{2}\right)^x y \) and \( 6^x = 2^x \cdot 3^x = y \left(\frac{3}{2}\right)^x y \).

Substituting these into the original equation:

\[
6 (2^{2x} \cdot 3^x) + 4 = 24 (2^x \cdot 3^x) + 2^x
\]

\[
6 \cdot y^2 \left(\frac{3}{2}\right)^x y + 4 = 24 y \left(\frac{3}{2}\right)^x y + y
\]

We will now simplify and solve this substituted form.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to explore a different question, or perhaps more details on the process? Here are some related questions:

1. How can we handle exponential equations with different bases?
2. What is the substitution method in solving equations?
3. Can logarithms be useful in solving this type of equation?
4. What are some typical strategies for solving equations involving exponential terms?
5. How does the change of variables technique work?
6. What are some common pitfalls when solving exponential equations?
7. How can we verify the solutions to such equations?
8. What are some alternative methods for finding the roots of exponential equations?

**Tip:** When dealing with equations involving different exponential bases, consider using logarithms or substitutions to simplify the expression.

Learn how to solve the exponential equation 6(12)^x + 4 = 24(6^x) + 2^x using the substitution method and logarithms. This problem involves different exponential bases and requires a systematic approach to find the solutions. Suitable for high school students and anyone interested in advanced algebraic equations.

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